ON FINDING THB LATITUDE AND LONGITUDE IN CLOUDY WEATHER AND AT OTHER TIMES, BY A. 0. JOHNSON, R.N. AUTHOR OP BOW TO FIND THE TIME AT SEA IN LESS THAN A MINUTE," dc. THIRTY-SIXTH EDITION. WITH NEW TIME-AZIMUTH AND EX-MERIDIAN TABLES. ALSO TABLES FOR FINDING THE LONGITUDE BY CHKONOMETEB, &o. (Supplied to H. M. Ships by Admiralty Order.) REVISED BY COMMANDER C. C. JOHNSON, R.N., lately Instructional Officer, H.M. Navigation School. PUBLISHED BY J. IX Admiralty Agent for Charts, 145, MINORJES ? > E-C. ^ ^ [ENTERED AT STATIONERS' HALL.] 1918. PRICE SIX SHILLINGS. LONDON PRINTED BY METCALFE & COOPER, LTD. GROCERS' HALL COURT, AND 18 &. 20, SCRUTTON STREET E.G. (2000.) , . ' I \ PREFACE TO THE 32ND EDITION. The publication of another edition of this little book affords the Author the opportunity of again expressing his thanks to the numerous officers of the Koyal Navy and Mercantile Marine who have from time to time favoured him with their opinions as to the value and utility of the following methods. Among those who, in the past, have so favoured him, he would gratefully mention the late CAPTAIN LECKY, E.N.E., for the prominence which he gave them in his "Wrinkles"; also CAPTAIN BLACKBUENE, of the P. and O. Service; and CAPTAIN OWEN, of the Union Line ; and the late Hydrographer of the Navy, by whom this book was ordered to be supplied to H. M. Ships. The accom- panying extracts will suffice to show the estimation in which it is held by practical navigators : The senior navigating officer of the squadron employed in towing out the Great Bermuda Dock says " During the passage I seldom got the sun at noon, and, had it not been for your Double Chronometer Method, I don't know what might have been the consequences, for we had hardly taken in our moorings when it came on to blow a most violent norther," &c. And an officer commanding a merchant vessel writes " My ship and another sailed at the same time from Liverpool, bound to Matamoras ; the weather being cloudy, I used your method, and arrived four days before the other ship, although she was a faster sailer ; and on my return from Pernambuco I did not see^the sun at noon for eleven days previously to making Cape Clear, but, trusting to my Double Chronometer, sighted the Cape just when I expected." This book has also been translated into French by Lieut. 0. V. de Jassaud ; German, by Theodor Liming of the Royal School of Navigation, Flensburg; Italian, by Captain Guarianti of the Italian Hydrographic Office, and Spanish by Captain Garci& Nunez of Santander, since awarded by the King of Spain the Eoyal Order of Naval Merit. It is also well known to American and Japanese Navigators, and a Turkish Version by Commander Mehmed Ali Bey, H.I.O.S., "Messoudieh," has recently been published. DARTMOUTH, 1909. NOTE. A Danish Translation was made by Robert Lundgren in 1912. A Russian version is in course of preparation (1917) by Captain V. Androunin, Transport "Mercury." 380028 OPINIONS OF THE PKESS, &c. Shipping and Mercantile Gazette. " It is expressed in such clear words, and the tables are so intelligible that they may be quickly understood by students. To the shipmaster this cheap and practical work, of a few pages, will be found a valuable assistant at sea." United Service Gazette. " Any simplification and condensation of the methods of finding the longitude at sea are very great desiderata. Mr. Johnson has conferred a great benefit on the nautical world, and deserves the gratitude of every navigator. The rules are simple and concise." From " Modern Navigation" by CAPTAIN HENRY TAYLOR, San Francisco, U.S.A., 1904. "... And as a final word we wish to state that there is no method in existence to-day of so much value to navigators as Johnson's." The Author has much pleasure in publishing one of many spontaneous tributes to the usefulness of this little book ; the more so, as it exactly describes the objects it is intended to accomplish : " I cannot refrain from expressing my admiration for 3' our little (?) work on ' Finding the Latitude and Longitude in Cloudy Weather.' This is not in any way due to the problem being new to me, as I have used it constantly for the last ten years. Possibly a rather cloudy and misty passage has emboldened an of ten -felt desire to tell you that the longer I know our little friend, the more I feel thankful for its tran- quilising effects, especially after a spell of S.W. winds in the ' Bay.' Only those in command can realise the comfort and pleasurable satis- faction it gives. If you .will kindly accept this assurance from one who is repeatedly deriving consolation from its use, it will gratify a long-felt wish," &c. CONTENTS. PAftK To Correct the Longitude for an Error in Latitude - 7 To Find the Longitude Simultaneously with the Noon Latitude - 9 Double Chronometer Method - 11 Time -Azimuth - 17 Ex-Meridian - 20 Hour Angle near Meridian 23 Combined Ex- Meridians - 23 Table I., for Longitude Correction, or Time-Azimuth - 26 Table II., Longitude Correction for Latitude and Bearing - 28 Table HA. For Double Chronometer Corrections by Inspection - 30 Table III., Ex-Meridian ... 32 Extension of Tables I. to III, to Latitude or Altitude 80 - 35 Table of Log Secants - 36 ,, ,, Half Log Haversines - 37 ,, ,, Log Haversines Hour-Angle 39 ,, ,, Log Cosines - - 40 Traverse Table, Altitude Correction, Arc into Time, &c. - 41 Examples Illustrating use of Tables - 43 Degree of Dependence - 44 Explanation of the Tables 45 Multiplication and Division by Tables I. and II. 47 Altitude -Azimuth Table 48 To Identify an Unknown Star - - 50 Tables for Finding the Stars - 51 Examples on Finding the Stars 52 Application of Tables I. and II. to Great Circle Sailing - 55 The Altitude -Azimuth by Tables I. and II. - 56 Construction of Table II. - 57 Principle of Double Chronometer Rule - 58 ,, ,, Time -Azimuth 59 ,, ,, Ex-Meridian 60 ,, ,, Altitude-Azimuth - 61 CORRECTING THE LONGITUDE FOR AN ERROR IN THE LATITUDE, AND ON FINDING THE LONGITUDE SIMULTANEOUSLY WITH THE LATITUDE AT NOON. It is frequently necessary for the officer entrusted with the navigation of a ship to calculate his longitude as soon as his sights are taken, in order to obtain without delay as close an approximation to the actual place of the ship as may be possible : but should only the latitude by dead reckoning be available, his result will be erroneous, unless, which is seldom the case, the dead reckoning be correct. Let us suppose that the navigator, on taking the sun at or near noon, has discovered that the latitude he employed is a certain number of miles in error ; he must then re-calculate his longitude with the corrected latitude, unless by any means he can correct that already found by making an allowance for the error in his latitude. To enable him to do this is the object of what follows : (I.) To find the Correction. From Table II. take the number corresponding to the latitude and bearing of the sun at the time of observation : this, multiplied by the correction for the latitude, will be the correction required. (II.) To name the Correction. Under the sun's bearing at the time of the observation write the opposite bearing, and suppose the letters to be connected diagonally, then that connected with the name of the correction for latitude will be the name of the correction for the longitude. 8 Thus, if the correction for the latitude were 10' N. and the sun's bearing S. 60 W., We should write down S.W. / And under it N.E. Then as the letter which stands diametrically opposite to N. (the name of the corr. for lat.) is W., the correction for longitude has to be allowed towards the West : and so on in other cases. The bearings may be taken from an Azimuth Table, or they may be easily found by means of Table I. In every case they are to be considered as less than 90; so that when the Tabular bearings exceed 90, we must subtract them from 180 , and reckon them from the opposite point of the compass ; thus, S. 120 W. would be N. 60 W., and so on. The correction may also be found by the Traverse Table, as follows : Enter the Table with the complement of the bearing as a course and the correction for the latitude as a diff. lat., and take out the corresponding departure. This converted into longitude in the usual way will be the correction required, and is to be applied as directed above. Example I. June 1. At 9 a.m., in lat. D.E. 52 10' N., when the sun bore S. 48 E., sights for longitude by chronometer placed the ship in 41 16' 5" W. At noon the above latitude was found to be 20 miles too far to the southward, and therefore the correction is 20' N. To find the true longitude (Tab. II.) Lat. 52, and bearing 48, give 1'.46; this number multiplied by 20 is the correction required. Approximate long 41 16' 45" W. S. E. Corr l'-46x20= 2912 E. / N. W. TruelongitudeatQa.m... 40 47 33 W. Here as the sun bore S. 48 E., we write down the letters S.E., and the opposite letters under them ; it is then seen that as the correction for the latitude is N., that for the longitude is E. Example II. July 25. The sun being obscured at noon at 2 p.m. in lat. D. K. 40 42' N., when the sun bore S. 64 W., sights for longitude placed the ship in 43 51' 45" E. At 9 p.m. by a star observation the above latitude was found to be 30 miles too far north the correction being accordingly 30 S. Find the true longitude. (Tab. II.) Lat 40 and bearing 64 give 0''64, which, multiplied by 30, is the correction required. Approximate long 43 51' 45" E. S. W. Corr O''64x30 = 19 12 E. \ N. E. True longitude at 6 p.m.. 44 10 57 E. In this case the bearing being S. 64 W., we write down the letters S.W. and the opposite letters under them ; it is then seen that as the correction for the latitude is S., that for the longitude will be E. To satisfy himself and to see what degree of depen- dence may be placed in the preceding rule, the navigator is recommended to put it to the test by his own observations. To find the longitude by observation at noon simultaneously with the latitude. Example III. At 8 a.m. sights for longitude worked with lat. D.E. 30 P 10' N. placed the ship in 20 12' W., and the sun bore S. 62 E. Steaming N.W. (true) 10' an hour, her lat. and long, brought up to noon by the log would be 30 38' N., and 20 45' W. But at noon the lat. by mer. alt. was found to be 30 48' N. Hence the correction for the above lat. was 10' N., and the correction from Tab. II. is -61 '. Therefore we have Approx. noon long 20 45' W. S. E. Corr '61x10'=: 6 E. / N.W. True long, at noon 20 39 W. Another advantage of the preceding method. When two or more men-o'-war are cruising in company, it is customary for each ship to show its latitude and longitude at about half an hour or so after noon. Any evolution that may be necessary is then executed. Now, if the preceding method were adopted, each ship could show her position at noon thereby saving valuable time, and possibly avoiding danger or inconvenience. 10 In cloudy weather the latitude may be found very expeditiously by means of Table III., which will give quite as satisfactory results as the more voluminous tables that are used for the purpose of reducing the altitude to the meridian at sea. The same by projection on the Chart. Through the position of the ship, as determined by the latitude D. E., and approximate longitude by observation, draw a line at right angles to the sun's bearing ; this is called the position line,' 1 " as the ship will be somewhere on it : the exact point will be where this line is cut by the parallel of the true latitude. In cloudy weather, when the ship is approaching the land, the position line produced will show its direction ; or if it runs parallel to the land will show the distance of the ship from it. Also, when in soundings, a cast of the lead will indicate approxi- mately the place of the ship on the position line. Examples for Practice. Time. Lat. Bearing. Cor. for Cor. for Lat. Long. Approx. Long. True Long. I A.M. 50 S. 60 E. 20 N. 18 E. 15 41 W. / 15 23 W. A.M. 40 S. 70 E. 10 S. 5 W. 1620E. 16 15 E. A.M. 20 S. 75 E. 15 N. 4 E. 17 50 W. 17 46 W. P.M. 60 N. 30 W. 16 S. 55 W. 40 13 W. 41 8 W. P.M. 45 S. 55 W. 18 S. 18 E. 345 W. 3 27 W. P.M. 10 N. 60 W. 25 N. 15 E. 445E. 5 OE. As a general rule, if the observations are taken at the ordinary times, the preceding method may be relied on, even though the error in latitude should amount to half a degree or more ; so that a re-calculation of the longtitude is quite unnecessary. The longitude found by a star observation may be corrected in the same way. * An easy way to find the direction of the position- line is to reverse the fiist letter of the bearing, and subtract the degrees from 90, thus if the sun bore S. 60 E., the direction of the position-line would be N. 30 E., or S. 30 W. DOUBLE CHKONOMETEK METHOD ; OR RULE FOR FINDING THE LATITUDE AND LONGITUDE BY TWO CHRONOMETER OBSERVATIONS, The ship's place may not only be determined with the utmost facility and accuracy by means of this rule, but it will also be found especially useful in cloudy weather, when there is little probability of being able to get an observation at noon. Every Navigator is familiar with the mode of finding the longitude by chronometer, and many careful ones make it their practice to take two observations at an interval of about an hour and a half or two hours. Those who do so may, with very little extra trouble, easily determine their latitude as well as their longitude, and thus be independent of the meridian altitude. The great utility of this method has been proved by the experience of officers both of the Navy and of the Merchant Service ; many of whom, by means of it, have made most successful passages across the Atlantic, to and from the West Indies, &c., when they have not been able to see the sun at noon for many days together.* RULE. I. Let two chronometer observations be taken at an interval of about an hour and a half or two hours, f and let the first be worked out with the lat. D.R. at the time of observation. II. Let the lat. D.R. and long, thus obtained be corrected for the run of the ship in the interval between the observations, and let the second observation be worked with this corrected latitude. Name these longitudes (1) and (2). III. The bearing of the sun at each observation is to be taken from an Azimuth Table. IV. Enter Table II, with the latitude and bearings, and take from it two numbers (a) and (b), of which take the difference, or sum, according as the bearings are in the same or adjacent quarters of the compass. J The difference of longitude divided by this difference or sum gives the correction for the second latitude ; and (a) and (b) multiplied by the correction for latitude give the corrections for the two longitudes. * It has moreover the advantage of being equally applicable to star obser- vations, which are daily assuming more importance in modern navigation. f Provided that the sun's bearing has changed not less than a point and a half, or two points, if possible. Vide p. 14. I Also, if the bearings are in opposite quarters, take the difference of (a) and (b). V. To apply the Corrections for the Longitude. When the observations are in the same or opposite quarter of the compass, Allow the corrections both to the East, or both to the West When the observations are in adjacent quarters of the com- pass, Correct the Easterly longi- tude towards the West, and the Westerly longitude towards the East in such a manner as to make the two longitudes agree. If they do not agree, they show that the corrections have been wrongly applied ; and herein we have a valuable safeguard against error, peculiar to this method only. VI. With either correction, and the corresponding bearing, find the name of the correction for the latitude, as in the preceding rule. Thus, suppose the correction for either longitude to be W., and the corresponding bearing S.W. : writing the letters N.E. under the above, we see that the letter opposite to W. is N., which is, accordingly, the name for the correction for latitude (2). Example I. March 7, at 8 a.m., in lat. D.R. 50 20' N., when the sun bore S. 60 E., chronometer sights placed the ship in 20 15' W. She then ran S.W. 20 miles, till 10 a.m., and her latitude being at this time 50 6' N., a second observation placed her in 20 56' W., the sun's bearing being S. 40 E. It is required to find the ship's true position at the time of the second observation. Bun S.W. 20 miles gives d. lat. 14' S., d. long. 22' W. Lat (1) 50 20' N. Long. 20 15' W. Ban 14 S. Ban 22 W. Lat. (2) BO 6 N. Long. (1) 20 37 W. Bearings (Tab. II.) Longitudes (a) (b) S. 60 E. -90 (a) (1) 20* 37' W. -90 185 8. 40 B. 1-85 (6) (2) 20 66 W. 20 20 Diflf. -96 95)1900(20 18-00 37'00 Long. (1) 2037'W. (2) 20 66' W. Lat. (2) 60 6'N. S. B. Correction 18 E. Corr. 37 E. Corr 20 N. / N. W. Long in 20 19 W. 20 19 W. Lat. in 50 26 N. The latitude of the ship is therefore 50 26' N., and the longitude 20 19 ' W. 13 If long. (2) confirms long. (1) it will be the true longitude of the ship, and show that lat. (2) is correct.* In the above example, both bearings being in the same quarter of the compass, we take the difference of (a) and (b) ; and, to avoid decimals, remove the decimal point two places to the right, both in divisor and dividend ; or, if preferred, proceed as directed on page 15. Example II. Oct. 10th, at 9 a.m. in lat. 40 N., when the sun bore S. 50 E., chronometer sights placed the ship in 20 40' E. ; she then ran N. 60 W. 30 miles, till 2 p.m., and her latitude being at this time 40 15' N., a second observation placed her in 20 26' E., the sun's bearing being S. 30 W. : it is required to find the ship's true place at the time of the second observation. Run N. 60 W. 30 miles gives d. lat. 15' N., d. long. 34' W. Bearings. S. 50 E. S. 30 W. (1) 20 Cor. Lat. Eun (1) 40 15 'N. N. N. 20 Long. Run Long. (1) Longitudes. (1) 20 6' E. (2) 20 26 E. 20 40' E. 34 W. 2 (b) 26 6 Lat. (2) 40 15 (Table II.) 1-09 (a) 2-26 (b) 20 6E. (a) 1-09 6 Sum 6'E. 6 E. 3-35 (2) Cor. 335)2000(6 26 'E. Lat. (2)40 14 W. Cor. 6-54 15'*N. 6 N. 13 S. / 56 E. / Long. 20 12 E. Long. 20 12 E. Lat. in 40 21 N. N.W. Hence the required latitude is 40 21' N., and long. 20 12' E. In this case we take the sum of (a) and (b), because the bearings are in adjacent quarters of the compass. The same by projection on the Chart. On the parallel of the second latitude lay down longitudes (1) and (2), and through these two points draw the corresponding position lines ; then where they intersect will be the position of the ship at the second observation. The position lines are found as directed on page 10. * In the same stars, Vj,de ,me way may be found the ship's position by the altitudes of two Lecky's " Wrinkles," Owen's " Stellar Navigation," &Q, Examplfs for Practice.* Observations in the same quarter of the compass. Diff. of a and b. Lat D.R. Bearings. Longitudes. Results. 50 ON. N. 60 E. N. 85 E. (1) 40 18 W. (2) 40 40 W. 50 29 N. 40 44 W. 48 20 N. S. 80 E. S. 51 E. (1) 20 15 E. (2) 19 45 E. 48 52 N. 20 23 E. Observations in adjacent quarters of the compass. Sum of a and b. Lat. D.R. Bearings. Longitudes. Results. 1 20 15 S. N. 40 E. N. 20 W. (1) 20 50 W. (2) 21 30 W. 1 20 6S. 21 2 W. 52 30 S. S. 80 E. N. 50 E. (1) 2 20 W. (2) 240W. 52 42 S. 224 W. The following are some results of actual observations taken in lat. 50 21' 30" N., and long. 3 34' 15" W. : Interval. Lat. used. Lat. by Oba. Long, by Obs. h. in. 5 8 1 18 GO H, 50 50 22 N. 50 21 48 N. O 1 II 3 35 45 W. 3 33 30 W. I 13 50 30 60 22 N. 3 34 W. 26 50 50 22 N. 3 35 I5W. 33 50 40 50 21 18 N. 3 33 45 W. 33. 50 50 21 54 N. 3 30 33 W. 54 50 50 23 N. 3 32 W. Several of these observations were taken under unfavourable conditions as to time and weather, the change in the bearings, in some cases, being very small, whereas it should not be less than a point and a half or two points, unless the observations be exceedingly good. The more nearly the bearings are at right angles to each other, the more accurate will be the results, and, as a general rule, the best results are given when the change in the bearings exceeds the lesser bearing. * In these examples the correction for run is supposed to have been applied as shown on pages 12 and 13. 15 NOTES. 1. The foregoing method is applicable not only to sun observations, which are of course to be preferred, but also to observations of two stars, or two observations of the moon or a planet, after a sufficient change of bearing. It may also be employed in the case of a sun observation taken before sunset and at an altitude of not less than 5 or 6, and the altitude of a star taken in the evening twilight; or of the moon or a planet, whichever is most convenient. In the case of a morning observa- tion it may be combined with a star taken before sunrise, and an observation of the sun taken an hour or two after, the proviso as regards the change of bearing being duly adhered to. 2. It will be noticed that the sum or difference of the two longitude corrections is equal to the difference of longitude. The sum when the bearings are in different quadrants, and the difference when they are in the same or opposite quadrants. This is another valuable check as regards accuracy. CALCULATION OF POSITION BY TABLE HA. This table has been inserted with the object of saving the trouble of multiplying or dividing, especially when the quantities are high numbers. In division, look for the divisor under " Nr. from Table II.," and the number to be divided in the same line, then at the top of the column in which it occurs will be the quotient. Thus, suppose we have to divide 19 by '95 as in Ex. I. p. 12, we look for -95 under " Nr. from Table II.," and 19 in the same line, then at the top of the column we find 20 ', the quotient required. In Ex. II. p. 13, we have to divide 20 by 3'35, which is the same as 10 by 1'67, which gives 6 and so on. It will be noticed that all three corrections are found in the same column, which is a great saving of time. N.B. If the number in the column " Nr. from Table II." does not exactly correspond with that from Table II. take the nearest or mean as the case may be, and if the error in lat. exceeds 31' enter with its half and double the result. This Table may also be used to save the calculation of the " run " for a portion of an hour by entering with the speed in the upper row of figures and the time in the margin. e.g. Speed 25 knots, to find run for 42 minutes. In the 25 ' column, on the same line as 42 minutes will be found the answer 17'5. 16 EXAMPLES ILLUSTRATING THE USE OF TABLE HA. Bearings. N. 60 E. N. 85 E. Example Table II. 90 14 Diff. -76 D I. Longitudes. (1) 40 18' W. (2) 40 40 W. . long. 22 Long. (1) 40 18' W. Corr. 26 W. Long. (2) 40 Corr. 40' W. 4 W. Lat. (2) 50 0' Corr. 29 N. N. Long, in 40 44 W. Long, in 40 44 W. Lat. in 50 29 N. CORRECTIONS. D. long. 22' -r -76 = 29' N. E. 29 x -90 =26 \ 29 x -14 = 4 S. W. To find the lat. corr. enter the Table with -76 (or -75) at the side and look for the nearest D. long. 21*7 in the same line, then at the top of the column will be found 29'. Example II. Bearings. Table II. Longitudes S. 70 E. -47 75 10' E. S. 50 W. 1-09 74 45 E. Sum 1-56 D. long. 25 Long. (1) 75 10 E. Long. (2) 74 45' E. Lat. (2) 40 20' N. Corr. 7$ W. Corr. 17 E. Corr. 16 S. Long, in 75 2$ E. Long, in 75 2J E. Lat. in 40 04 N. CORRECTIONS. D long. 25' ~ 1-56 16' * nearly S. E. 16 x -47 = 7J \ 16 x 1-09 - 17$ N. W. NOTE. If the divisor for the diff. long, exceed the limit of the Table, divide it by 2 or 3, as the case requires ; also divide the diff. long, by the same number, and proceed as before. * Here 1'55 in the side column and 24-8 (or 25) in the same line give 16' at the top. i 17 TO FIND THE LONGITUDE-CORRECTION AND TIME-AZIMUTH BY TABLES I. AND II. NOTE. These Tables can be adapted to any latitude between 60 and 80 by means of the Supplementary Tables on page 35. TO FIND THE TIME-AZIMUTH. Take from Table I. the numbers for the H.A. and Lat., and for the H.A. and Dec. The sum or difference of these in the proper Latitude Column of Table II. gives the Bearing, or Azimuth, which will be found on the left-hand side of the Table.* TO NAME THE AZIMUTH. Mark the first number with the opposite name to the Lat., and the second with the same name as the Dec. When the names are the same, take the sum with the common name; when different take the difference with the name of the greater. This will be the point from which to reckon the Azimuth. Exception. When the H.A. exceeds six hours, mark the first number with the same name as the Lat. and proceed as before. Examples. 1. Lat. 40 N., Dec. 20 N., H.A. 3h. 48m. E. of Mer. For Lat. 40 N. and H.A. 3h. 48m. we have '54 S. Dec. 20 N. and H.A. 3h. 48m. we have "43 N. Diff. -11 S. By calculation S. 85'l E. 2. Lat. 20 N., Dec. 14 S., H.A. 4h. 40m. W. of Mer. The above give -13 S. And -27 S. Sum -40 S = S. 69 W. Table II. By calculation S. 69 e> 5 W. 3. Lat. 46 N., Dec. 14 N., H.A. 6h. 40m. E. of Mer. For the above we have ... ... ... -18 N. And -26 N. Sum -44 N. = N. 73 E, Table II. By calculation N. 73 2 E As the H.A. exceeds 6 hours we subtract it from 12 hours and enter the Table with the remainder, or 5h. 20m. N.B. The sum or difference found as above is also the correction in longitude for 1' error in the latitude, so that two important elements are found simultaneously. * The numbers for intermediate degrees and hour-angles are easily taken out at sight and with sufficient accuracy for all purposes for which these Tables are intended vide page 19. 18 FOB HOUK-ANGLES LESS THAN AN HOIJE. When the Hour-angle is less than' one hour, find the azimuth for one hour and multiply it by the minutes expressed as the decimal of an hour. Example. Lat. 32 N., Dec. 12 N., H.A. Oh. 36m. E. of Her. For the above we have 2' '33 S. And -79 N. Diff. 1-54 S. = S. 37 E. Table II. /. The Bearing at Oh. 36m. =37 x -6-22 or S. 22 E. This is true, nearly, because, near the meridian, the azimuth varies as the Hour-angle, approximately. NOTES. I. In actual practice it will generally be sufficient to take from Table II. the Bearing which most nearly agrees with the sum or difference ; or the Mean Bearing, as the case may be. II. If we wish to find it more exactly, we take the diff . of the sum or diff. and the first of the two numbers between which it lies, also the diff. of these latter, and make a fraction with the two differences. This fraction multiplied by 2 or 120', gives the number of minutes to be added to the first of the two bearings. Vide Examples (c) and (d) page 19. III. The H.A. for the Lat. is on the left-hand side of Table I., that for the Dec. on the right. It will be noticed that the intervals in the latter are greater than those in the former; but as the numbers corresponding to them change very slowly it will suffice to take the nearest H.A. to that given, or the mean of the two between which it lies. Thus for 3h. 4m. we should take 3h. 6m. ; but for 3h. 16m. we should take the mean between 3h. 6m. and 3h. 26m. and so on. The H.A.'s for the latitude may be taken out to the nearest 4m. by taking the means ; and the latitude and declination to the nearest degree in like manner. The easiest way to take the means is to add half the difference of the two numbers to the lesser number, which can readily be done at sight. \Vhen the H.A. lies between two of those given and the latitude or declination consists of an odd number of degrees, proceed as in the following 19 Examples. (a) Lat. 43 N., H.A. 2h. 4m. We have Lat. 42 and 2h. Om. = 1-56' 44 ' 2h. 8m. = 1'54' Means : 43 2h. 4m. = 1-55' (b) Dec. 15, H.A. 2h. llm. We have Dec. 14 and H.A. 2h. 8m. = -47' 16 2h. 14m. = -52' Means: 15 2h. llm. = -49'. The above are easily taken out at sight. EXPLANATION OF TABLE II. The construction of this Table is fully explained on page 57. When the latitude and bearing are given to find the correction it is taken out at sight. Thus for Lat. 50 and Bearing 60 we have '90 50 61 -86 51 61 -88 In the latter we take the mean of 50 & 60, 52 & 62, the two even degrees next less and the two even degrees next greater than those given. Conversely : For Lat. 50 and corr. 90' the bearing is 60 50 86' 61 51 88' 61 The bearings found in this manner will generally be within a few minutes of those obtained by calculation, and sufficiently accurate for laying off position lines or finding the compass correction, &c. When still greater accuracy is desired proceed as follows : (c) To find the bearing for lat. 40 and l'-15 ' Diff. We have 1-151 o 4 * 1-17 f /. x 2 = = = 30' 1-09} 8 8 8 2 which, being added to the lesser bearing we have 48 30 ', vide below (d) To find the bearing for lat. 41 and l/'Sl ' Diff. We have 1-31 ' 4 . 1-35 j /. x 2 = - = 1 --= 136' 1-30| 5 5 55 which, being added to the lesser bearing we have 45 36' In Ex. (c) 1-17 and 1'09 are the numbers in Lat. 40 column 20 between which 1-15 lies; and in (d) the Lat. being 41 the numbers are taken from the columns for Lat. 10 and 42. The name of the correction for longitude is found as directed on p. 8, or by reversing one of the letters of the Bearing : thus if the body bore S.E. we should have N.E., or S.W. denoting that corrections N. and E. go together, as also S. and \V. TABLE II. is used also for finding THE EKEOK IN LONGITUDE DUE TO 1' EKEOK IN ALTITUDE WHEN THE BEARING is KNOWN. For this purpose we take the nearest or mean Bearing from the last column but one of the table, then with the latitude and this Bearing take out the correction as before. Thus: for Lat. 40 P and Bearing 46, the correction is l'-80; and for Lat. 34 and Bearing 69 it is 1/-29, the mean between l'-34 and l'-25 the numbers which correspond to 65 and 74, between which Bearings 69 lies, and so on. From lat. to 34 the correction is taken from the first page of Table II. TO NAME THE COKKECTION. When the observed altitude is too small the correction takes the same name as the Bearing. Thus, if the body bore S.E., the correction would be East; or if S.W., it would be West. When too large it takes the opposite name. Or, we could multiply the above correction by 4 to convert it into seconds, and apply it to the Hour-angle, subtracting if the observed altitude is too small and adding if too great. The Table also shows the degree of dependence in an observation, a? far as the altitude is concerned. Thus in the above example it is seen that each minute of error in the altitude produces an error of l'-80 in the longitude or of 7*20 sec. in the Hour-angle. TABLE III. THE EX-MERIDIAN TABLE. With the latitude and altitude take out N. ; then with N. and the B..A. find the Reduction. Example : Lat. 50, Alt. 40, H.A. Oh 15m (1) Lat. 50 and Alt. 40 give -85 for N. (2) N. -85 and H.A. 0-15 give 6'2 or 6'-12". This added to the true altitude gives the Mer. Alt. 21 The Latitude is then found by the Mer. Alt. Eule. When the declination is small its effect upon the Eeduction may be neglected; but when considerable a second correction, always subtractive, may be taken from the small table annexed. Example II. Lat. 50 N., Alt. GO , Dec. 20 N., H.A. 18m. 30a. (1) For lat. 50 and alt. 60 N. we have 1-29 (2) For N. 1-29 (or 1-30) and 18m. 30s. Bed. = 14' -5 (3) For Dec. 20 and Red. 14-5 (or 15) Corr. = -g /. The true Reduction = 13-6 or 13' 36" which, added* to the true alt., gives the Mer. Alt. For 18m. 30s. we take the mean of the numbers for 18m. and 19m. ; or multiply the difference of the numbers for 18m. and 19m. by '5 and add to the lesser. In this way we can take out the Eeduction for any number of seconds in the H.A., for we have only to multiply the diff. by the number of seconds, expressed as the decimal of a minute, and add the result to the lesser number as before. The values of the Eeduction are tabulated to 35m. which will probably be found sufficient for most purposes. Should, however, the H.A. be greater than this, take out the Eeduction for its half and multiply it by 4. Thus, for 40m. 24s., whose half is 20m. 12s., or 20'2m. and N. T30, We have N. 1-30 and 20m. = 17 Diff. 1-8 x -2 = 0-4 17-4 4 .-. Reduction for 40m. 24s. = 69-6 If in the above the dec. were 20, we should have as before 69-G Dec. 20 and 70' = 4-2 .*. The true Reduction = 65-4 Among other advantages, the above little table shows at a glance the value of the reduction for any number of minutes within the limits tabulated, and the effect that an error of a given number of minutes in the H.A. would produce on the resulting latitude, and therefore the degree of dependence. * In observations near the Meridian below the pole, the Reduction is to be subtracted, instead of added. 22 EXTENSION OF TABLES I. III. TO LAT. 80 N. or S. T^ I. When the latitude exceeds 58, take the equivalent latitude from Tab. (i.), and proceed as follows : Example I. Lat. 70 N., Dec. 20 N., H.A. 2h. 8m. Tab. (i.) Tab. (I.) Lat 70 = 54 I Lat. 54 and H.A. 2h. 8m. = 2'-20x2 = 4'40 Multr. =2 ) Dec. 20 and H.A. 2h. 8m ....... - '68 Diff ............. = 3-72 .-. Long. Corr. = 3-72. . II. When the latitude exceeds 60, take the equivalent latitude from Tab. (ii.), and with this latitude and the bearing take out the Long. Corr. as before, and multiply it by the number under the latitude. Example II. Lat. 74, Bearing 60. Tab. (ii.) Tab. (II.) Lat> 56 and B ^ aring 60 = 1<03 . Long. Corr. = 1-03 x 2 = 2-06. Conversely : Given Lat. 74 and Long. Corr. 2-06, to find the Bearing. We have 2'06 -5- 2 = 1-03, and Lat. 74 = Lat. 56. .-. Lat. 56 and 1-03 - Bearing 60. EXTENSION OF EX-MER. TABLES TO LAT. OE ALT. 80. When the latitude is greater than 60, take the equivalent latitude from Tab. (iii.), and find N. for this latitude, and the given altitude. With N. and the H.A. find the reduction as before, and divide by the number standing under the latitude in Tab. (iii.) Example I. Lat. 74 N., Alt. 12, Dec. 4 S., H.A. 16m. 6s. Tab. (iii.) Tab. (III.) Lat. 74 = 56 ( Lat. 56 * ... , N . Divisor = 2 ! Alt. 12 \ * n \** N. -57 and 16m. 4s. = 4'-8 .. The reduction = i^ 8 = 2' -4 = 2' 24". When the altitude exceeds 60, take the equivalent altitude from Tab. (iii.) and with this and the latitude take out N. Find the Reduction as before and multiply it by the number standing under the Alt. in Tab. (iii.) Examvle II. Alt. 72, Lat. 10 N., H.A. 10m. Tab. (iii.) Tab. (III.) Alt. 72 = 52 I Lat. 10 I ,., / N , Multr. 2 5 Alt. 52! - 15 9< N -) N. 1-59 and H.A. 10m. = 5'-3 .. The Reduction - 5 '"3 x 2 10' -6 or 10' 30*. 23 THE TEUE ALTITUDE NEAE THE MEEIDIAN AND THE MEEIDIAN ALTITUDE BEING KNOWN, TO FIND THE H.A., NEAELY. Example I. Lat. 50 N., Alt. near Her. 39 53' 48", Mer. Alt. 40 0' 0" : to find H.A., approximately. Lat. 50) _ .* rNM Mer. Alt. 40 0' 0" Alt. 40 j ~ Alt. nr. Mer. 39 53 48 Diff. ... = 6 12 ( = 6' -2) .-. N. -85 and 6' -2 = 15m. Os., the H.A. required. Example II. Lat. 50 N., Alt. 60, Dec. 20 N., Mer. Alt. 60 0' 0" Alt. near Mer. 59 46 24 13 36 = 13'-6 Lat. 50 J ) i no. TJ Dec. 20 & 15' = +'9 Alt. 60j : 31^5 N. 1-29 and 14'5 = 18m. 28s. To obtain this, we see that, in the line for N. 1'30, 14-5 lies between 13'8 and 15'3 the numbers for 18m. and 19m. Diff. .*. We have given nr. 14*5 ) 18m. = 13-8 [ .'. TZ x 60" =28 sec. 19m. = 15-3[ .-. The H.A. required is 18m. 28 sec. Both observations must be accurately taken and the first corrected for the run in the interval. This method may be employed when the Sun has been obscured till it is too late to take the usual observations for time. When the Ship time is not known with any degree of certainty a second ex-meridian should if possible be taken in the afternoon at about the same altitude as the first, and the mean of the two latitudes (reduced to noon) may be taken as the true latitude. By this means any errors in the reductions due to errors in the time are eliminated. COMBINED EX-MEEIDIANS. When a second observation is taken on the same side of the meridian, and the second latitude confirms the first, it may be assumed to be the true latitude. But, if not, take the difference of the two latitudes, multiply it by the lesser H.A., divide by the elapsed time, *and apply the result * Or multiply the Diff. Lat. by the Nr. taken from Tab. (iv.), p. 35. 24 to the latitude given by the observation nearest to the meridian, adding or subtracting according as this latitude is greater or less than the other. When the observations are on opposite sides of the meridian also apply the Correction to the lat. given by the observation nearest to the meridian, but the opposite way, i.e., subtracting or adding according as this latitude is greater or less than the other. Both observations should be accurately taken and the second corrected for the run in the interval. Example. Lat. DR. 50 0' N. 1st. Obs., H.A. 23m. 30s. Alt. 39 39'. 2nd. H.A. 8m. 20s. Alt. 39 56'. Obs. Interval H.A's. 23 30 8 20 15 10 *1 Alt. 89 39' Bed 4- 15 89 54 Z.D. 50 Dec. J) Lat. (1) 50 Correction 4 x 8-3 15-2 - 2 (or by Tab. iv.) corr. = 4' x -53 = 2' to the nearest minute) 2nd Alt. 39 56' Red. + 2 39 58 Z.D. 50~2~ Deo. 4 S. Lat. (2) 49 58 N. Corr. 2 Lat. in 49 56 N. The Diff. Lat., 4', multiplied by the lesser H.A. and divided by the interval gives 2' (about) which is subtracted 1 from Lat. (2) because this Lat. is less than the other. The latitude thus found is for the time of the second observation. Example for Practice : The Azimuth. Lat. it. 40 N Dec. 10 N. H A. 3-16 E. Ans. Az. S. 69 B. 30 S. 12 S. 4-20 W. M N. 88 W. 22 N. 20 S. 3-48 E. ( S 57 B. 32 S. 14 N. 4-12 W. N 63 W. 50 N. 20 N. 7-16 E. H M N 63 E. 58 S. 24 S. 6-52 W. S. 66 W. 44 N. 56 N. 5-30 E. ,, N. 45 E. The Reduction to the Meridian. Alt 40 Dec. H A. 10m. 5s. Ans . Red. + 38 12 S. , 14m. 30s. j _j_ 60 10 S. , 16m. 24s. + 1 50 20 N. , 17m. 48s. 42 22 N. , 12m. 3s. D _l_ ij 52' 12 N. 32m. 12s. 1 i +8 2' 48" 6' 36" 6' 36" 7' 24" 1' 54" LONGITUDE CORRECTION, TIME-AZIMUTH, AND EX-MERIDIAN TABLES 26 TABLE I. FOB THE LONGITUDE COBBECTION, LATITUDE o o o o o C3 o o o o o o o o C) HA a 2 4 6 8 10 12 14 16 18 20 22 24 26 28 ^ IT for HA. fn-n Lat O'OO o-oo 0-03 O'O7 O'lO O'l^ 0-18 0'2I 0-25 0-29 0'33 0-36 0*40 0-45 0*49 o'S3 IOr Dec. H,M H.M. 1.0 373 O'OO 0-13 0-26 0-39 0-52 0-66 079 0-93 1-07 I'2I 36 51 66 1-83 1-98 1.0 4 3'49 O'OO 0'12 0-24 0-37 0'49 0-6 1 074 0-87 I '00 I'lj 27 41 '55 170 1-85 8 8 3"27 0'00 O'll 0-23 0-34 0-46 0-58 0-69 0-82 0'94 I '06 19 32 46 i'59 174 12 12 3'08 O'OO O'll 0'22 0-32 o*43 0-54 0-65 077 0-88 I'OO * I 2 24 '37 i'50 1-64 16 16 2-90 O'OO O'lO 0'20 0-30 0-41 0-51 0-62 072 0-83 0'94 c6 17 29 1-42 i'54 20 20 275 O'OO 0-09 0-19 0-28 0-39 0-48 0-58 0-68 079 0-89 oo II '22 1-31 1-46 24 24 2-60 o-oo 0-09 0-18 0*27 0-37 0-46 o'55 0-65 075 0-85 O'95 08 16 1-27 1-38 7O 28 2.47 o-oo O'o8 0-17 O'26 o'35 0-44 0-53 0-62 071 0-80 0-90 '00 10 I'2I 1-32 ? 36 32 2-36 O'OO 0-08 0-16 0-25 0-33 0-41 o'5o 0-59 0-68 076 0-85 o'95 05 I'I5 1-25 40 36 2-25 o-oo 0-08 0-16 0-24 0-32 0-40 0-48 0-56 0-64 073 0-81 0.91 I'OO 1-09 1-19 46 40 2-14 O'OO 0-07 0-15 0-22 0-30 0-38 O'4O 053 o'6i 070 078 0-87 0'95 i '05 1-14 C2 44 2-05 o-oo 0-07 0-14 0'2I 0-29 0*36 0'44 0-51 0-59 0-67 075 0-83 0-91 I'OO 1-09 56 48 1.96 O'OO 0-07 0-14 0'2I 0-28 0-35 0*42 49 0-56 0-64 071 079 0-87 0-96 1*04 2.2 52 88 O'OO 0-06 0-13 0'20 0-26 o'33 O'4O 0-47 0-54 0-61 0-68 076 0-84 0-92 I'OO 8 56 80 O'OO 0-06 0-13 0-19 0-25 0-32 0-38 0'45 0-52 o ! 59 0-66 o73 0-80 0-88 0-96 14 2.0 73 O'OO 0-06 0'12 O'i8 0-24 0-30 o-37 o'43 0-50 0-56 0-63 070 077 0-84 0-92 20 8 60 o-oo 0-06 O'll 0-17 0'22 0-28 0-35 0-40 0-46 0-52 0-58 0-65 071 078 0-85 34 16 48 O'OO 0-05 O'lO 0-16 0'2I 0-26 0-31 0-37 O'42 0-48 0'54 o'6o 0-66 072 079 48 24 38 O'OO 0-05 O'lO 0-14 0-19 0-24 0-29 0-34 0'39 0-45 0-50 0-56 0.61 0-67 073 3.6 32 28 O'OO 0-04 0-09 0-13 0-18 0-23 0-27 0-32 Q'37 O'42 0-47 0-52 0'57 0*62 0-68 26 40 19 O'OO 0-04 0-08 0'12 0-17 0'2I 0-25 0-30 0'34 0'39 0'43 0-48 o'53 Q'58 0-63 48 48 i i O'OO 0.04 0-08 0'12 0-16 O'2O 0-24 0-28 0*32 0-36 0-40 o'4S 0-49 0-54 0'59 4.16 o 5 ! 03 O'OO 0*04 0-07 O'll 0-15 O'i9 O'22 0-26 o'3O 0-34 0-38 0-42 0-46 0-50 0'55 5.4 3.0 oo O'OO 0-04 0-07 O'lO 0-14 0-18 0'2I 0-25 O'29 0-32 0-36 0-40 Q'44 0'49 0'53 6.0 '8 0-93 O'OO 0-03 0-06 o'io 0*13 0-16 0-19 0*23 0-27 0-30 0.34 0-38 0-41 0'45 0-50 16 0-87 O'OO 0-03 0-06 0-09 0'12 0-15 0-18 O'22 0-25 0-28 0-32 o-3S 0-39 0-42 0-46 24 0-81 O'OO 0-03 0-06 0-08 O'll 0-14 0-17 O'2O 0-23 O'26 0-29 o'33 0-36 0-39 o-43 32 075 O'OO 0-03 0-05 0-08 O'll 0-13 0-16 0-19 0'22 0-24 0-27 0-30 Q'34 0'37 0-40 40 070 O'OO 0'02 0-05 0-07 O'lO O'I2 0-15 0-17 0'20 0-23 0-25 0-28 0-31 0'34 o*37 48 0-65 000 O'O2 0-04 0-07 0^09 O'll 0-14 0-16 0*19 O'2I 0-24 0-26 0-29 0-32 o'34 * 56 0-60 O'OO O'O2 0-04 0-06 O'o8 O'lO 0*13 0-15 0'I7 0-19 O'22 0-24 0-27 0-29 0-32 EC 4.0 0-58 O'OO 0'02 0-04 O'o6 O'o8 O'lO 0'12 0-14 0'I 7 0-19 0'2I 0-23 O'26 0'28 0-31 | 8 o-53 O'OO 0'02 0-04 0-06 0-07 0-09 O'll o'i3 0-I 5 0-17 0'19 0'2I O'24 0-26 0-28 5* 16 0-49 O'OO 0'02 0-03 0-05 o'O7 o'O9 O'lO O'I2 O'I4 0-16 0-18 O'2O 0'22 0-24 O'26 I 24 0-44 O'OO O'O2 0-03 0-05 0-06 0-08 0-09 O'll 0'13 0-14 O'i6 O'l8 O'2O O'22 0-24 5T 32 0-40 O'OO O'OI 0-03 0-04 O'Ob 0-07 0-09 O'lO 0'12 0-13 0-15 0-16 O'l8 0'20 O'2I 3* 40 0-36 O'OO O'OI 0-03 0-04 0-05 O'o6 0-08 0-09 O'lO 0'12 0-13 0-15 0-16 0-18 0'19 ** 48 0-32 O'OO O'OI O'O2 0-03 0-05 O'o6 0-07 0-08 0'09 O'll O'I2 0-13 0-14 0-16 0'17 ? 56 0-29 O'OO O'OI 0'02 0-03 0-04 0-05 0-06 0-07 0-08 0'09 O'lO 0'12 0-13 0-14 0'15 0^ 5.0 0-27 O'OO O OI O'OI 0-03 0-04 0-05 0-06 0-07 0-08 0'09 O'lO O'll 0'12 0-13 0'14 5' P 8 0-23 O'OO O'OI O.OI O'O2 '3 0*04 0-05 0-06 0-07 0-07 0-08 0-09 O'lO O'll 0'12 t*- 16 0-19 O'OO O'OI O'OI O'O2 0-03 0-03 0-04 0-05 0*06 0'06 0-07 0-08 0-09 O'O9 O'lO 24 0-16 O'OO O'OI O'OI 0'02 0-03 0-03 0*03 0-04 0-04 O'OS 0-06 0-06 0-07 0-08 0-08 32 O'I2 O'OO O'OI O'OI O'OI O'O2 O'O2 0-03 o 03 0-03 0-04 0-04 0-05 0-05 0-06 0-06 40 0-09 O'OO 0.01 O'OI O'OI O'OI 002 O'O2 002 O'O2 0-03 003 0-03 0*04 0-04 0-05 5 0-04 O'OO O'OO TOO O'OO O'OO O'OI O'OI O'OI O'OI O'OI O'O2 0'O2 0'02 0'02 0'02 6.0 O'OO o oo C'C-0 D'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO O'OO o ~^~ g o o o o O o n o 2 4 10 12 14 16 18 20 22 24 26 28 * DECLINATION For hour-angles less than an hour, vide page 18. 27 TABLE I. AND THENCE THE TlME-AziMUTH BY TABLE II. LATITUDE o o O O o O o o o o o o o HA. a 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 H- for H.A. Lat. ^ ^^ ^ MMHM . .^ .^ _ for O'OO 0-58 0-63 0-67 073 078 0-84 0'90 0-97 1-04 1*11 1-19 1-28 i'35 1-48 i -60 Dec. M.M. H.M. 1.0 373 2-15 2-33 2-52 271 2-92 3-13 3-36 3-60 3-86 4-14 4'45 478 5-14 5'53 5'97 1.0 4 3'49 2'01 2-18 2-35 2'53 271 2-93 3'14 3'37 3-61 3'87 4-15 4-46 4-80 5-i7 8 8 3-27 I-8 9 2 'O4 2'2I 2-38 2 '55 274 2'94 3-16 3'39 3' 6 3 4-19 4'5 4-85 5-23 12 12 3'o8 78 1-92 2 '08 2-24 2-40 2-58 2'77 2-97 3;i9 3'42 3-67 3'94 4-24 4-56 4-92 16 16 2'9O 68 r8i I ' 9 6 2' 1 1 2-27 2-44 2'6l 2-80 3-22 3]46 372 4-00 4-31 4-65 20 20 275 59 172 1-85 2'00 2-15 2-30 2'47 2-65 2-84 3-05 3-52 378 4-07 4-40 24 24 2-60 'So 1-63 176 I-8 9 2-04 2-19 2'35 2-52 270 2-89 3'io 3]33 3'59 3-86 4-17 30 28 2-47 43 *'55 I'6 7 r8o 93 2-08 2-23 2'39 2'55 275 2 '95 3'4i 3-67 3'96 36 32 2-36 36 1-47 i'S9 171 84 1-98 2'12 2-27 2'44 2-62 2'8l 3-01 3-24 3'49 377 40 36 2-25 30 1-40 ''5 1 1-63 75 r88 2'O2 2-17 2'33 2-49 2-68 2-87 3-09 3'33 3'59 4 6 40 44 2-14 2-05 24 18 i '34 1-28 1-56 1-49 67 60 r8o 172 :n 2-07 1-98 2'22 2'12 2-38 2-28 2-56 2-44 274 2-62 2'95 2-82 3-04 3'43 3-28 5 2 56 48 96 13 T22 1-32 i '43 53 1-65 77 1-89 2-03 2-18 2'34 2-51 270 2-91 3-14 2.2 52 88 09 I'I7 1-27 i'37 '47 1-58 69 1-82 2-09 2-24 2-41 2'59 279 3*01 8 56 80 04 I-I3 I'22 1-31 41 62 174 1-87 2'00 2-15 2-31 2-48 2-67 2-89 14 2.0 73 roo ro8 1-17 1-26 '35 i '45 56 1-67 79 '92 2-06 2'22 2-38 2'57 277 20 8 6c 0-92 roo ro8 1-16 '25 i*34 '44 66 78 91 2-05 2'20 2'37 2-56 34 16 48 0-86 0-93 roo ro8 16 1-24 '33 i'43 53 65 77 r90 2'O4 2'2O 2'37 48 24 38 079 0-86 0-93 roo 07 I1 5 24 i'33 42 53 64 76 1-84 2'04 2'20 3.6 32 28 074 0-80 0-86 Q'93 oo 1-07 15 1-24 32 42 52 6 4 I' 7 8 '90 2'OS 26 40 19 0-69 074 0-83 0-87 0-93 roo 07 23 .32 42 52 1.64 77 I'9I 48 48 ii 0-64 0-69 075 0-81 0-87 0-93 roo 1-07 * ! 5 23 32 '42 ! '53 65 78 1.16 56 03 0-60 0-65 070 075 0-81 0-87 0-93 roo 07 it 23 32 1-42 '53 66 5.4 3.0 '00 0-58 0*62 0-67 073 078 0-84 0-90 0-97 04 i i 19 '28 48 .60 6.0 8 0'93 0'54 0-58 0-63 0-68 073 078 0-84 0'91 0-97 1-04 ii 19 1-28 38 '49 =- 16 0-87 0-50 o'54 "59 0-63 0-68 073 078 0-84 0.90 0-96 04 'II I'2O 29 38 24 0-81 o'47 0-51 o'SS o'59 0-63 0-68 073 078 0-84 0-90 0-96 04 I'll 20 30 32 075 '43 0-47 0-51 0-55 '59 0-63 0-68 073 078 0-84 0-90 0-96 1-04 12 21 40 070 0*40 o'44 0-47 0-51 '55 '59 0-63 0-68 072 078 0-83 0*90 0-96 04 12 48 0-65 o*37 0-41 0-44 0'47 0-51 0-54 0-58 0-63 0-67 072 077 0-83 0-89 0'96 04 B 56 O'6o o'35 0'37 0*40 Q'44 0-47 0-50 Q'54 0-58 0-62 0-67 071 077 0-83 0'89 0-96 33 4.0 0-58 0-36 0-39 0-42 0-45 0-48 0-52 0-56 O'6o 0-64 0-69 074 079 0-86 O'92 8 0'53 0-31 0'33 0-36 0*39 0-41 0*45 0-48 0-51 0-55 0'59 0-63 0-68 073 079 0'85 16 0-49 0-28 0-30 Q'33 0'35 0-38 0-41 0'44 0-47 0-50 o'54 0-58 0-62 0-67 072 078 3 24 0-44 O'26 0-28 0*30 0-32 0-37 0-40 0'43 0-46 0-49 o-53 0-61 0-66 071 001 32 O'4O 0-23 0-25 0-27 0-29 0-32 0-36 0-39 0*42 0*45 0*48 0-52 0-56 0-60 0'6$ 2 40 0-36 O'2I 0*23 0-25 0-26 0-28 0-30 0'33 o*35 0-38 0*40 o-43 0-47 0*50 0'54 0-58 48 0-32 O'i9 O'2O 0-22 0-24 0*25 0-27 0-29 0-31 0'34 0-36 o-39 0-42 0*45 0-48 0-52 4 587 6ri 63-4 65-8 68-1 7*5 72.9 235 2 40 40-8 43-2 48-0 52-8 5 5 '2 57-6 60-0 62-4 64-8 67-2 69-6 72-0 74-4 240 32 TABLE III. EX-MERIDIAN TABLE. LATITUDE. Alt. 6 4 8 10 12 14 16 18 20 22 o 24 26 28 30 32 34 N N N N N N N N N N N N N N N N 8 I'OI 01 TOO I'OO 0'99 0'98 O'97 0'96 '95 0*94 0*92 O'9I 0*89 0-88 0-86 0-84 10 I'OI ol I'OO I'OO 0-98 o % 97 0-96 0'95 '94 0-92 o'9i 0-89 0-88 0-86 0-8 4 12 1-02 02 I'OI roi I'OO 0-99 0-98 0'97 0'95 o'93 0-91 0'90 0-88 0-87 14 16 1-03 1-04 3 '04 I '02 1-03 I'02 1-03 oi 02 TOO roi I'OO 098 0-98 o'-97 0-94 0-95 0-92 0-94 0-92 0-89 0-90 0-87 o'88 0-86 18 1-05 05 1-04 I'04 03 I '02 I'OI I'OO 0-99 0-98 0-96 0-95 0'93 0*91 0-89 0-87 20 22 I -06 1-07 I '06 1-07 ro6 Ml 04 05 1-03 1-04 I '02 1-03 I'OI 1-02 I'OO I'OI I'OO 097 0-98 0-96 0-97 o ; 95 0-92 0-93 0*90 0-91 0-88 0-89 24 1-09 1-09 i -08 i -08 07 I -06 1-05 1-04 I'02 I'OI I '00 0-98 0-96 0-94 0*92 0*90 26 I'll rn no 1-09 08 I -08 1-07 I '06 I'04 03 I'OI oo 0-98 0-96 0-94 0-92 28 I'I3 1-13 1*12 I'll 1C no 1-09 I -08 I -06 05 1-03 02 I '00 0-98 0*96 0-94 30 1*15 ' 15 1-14 ''3 12 I-I2 I'll I'lO ro8 07 1-05 04 I'02 1*00 0-98 0^96 32 n8 riS 1-17 ri6 15 1-14 1-13 I'12 I'll 10 c8 06 1-04 'O2 I'OO 0-98 34 I'2I 1*21 I*2O 1-19 18 n6 1-16 I'I5 1*13 "12 *IO 08 1-06 'OS I '02 I'OO 36 I'24 1-24 1-23 I'22 21 1-20 1-19 n8 ri6 '5 13 'II 1-09 '07 I -06 1-03 38 T27 1-27 T26 1-25 24 T23 1*22 1*21 1-19 18 16 '14 ri2 10 I -08 06 40 I'3I I-3I I'3O 28 1-27 I'26 1-25 1-23 21 20 18 n6 14 II 09 42 '"35 i'35 i'34 i'33 32 I-3I I'30 I'29 1-27 '25 23 21 1-19 17 '14 12 44 i'39 i '39 1-38 1-37 36 ,.35 I'34 i'33 i'3" 29 27 25 1-23 '21 18 'IS 46 I -44 1-44 i '43 1-42 41 I'40 1-38 i'37 ''35 '33 '3' 29 1-27 25 22 19 48 1-49 .1-49 1-48 1-47 4 6 '45 1-42 1-40 38 36 '34 1-32 29 26 24 50 i'55 i'55 1*54 i'53 52 1-51 i'49 1-48 1-46 '44 '42 40 i*37 '34 31 29 52 1-62 1-62 i - 6o i59 '58 1-57 1-56 J'54 1-52 'SO 48 46 i'43 4 '37 '34 54 170 170 1-68 r68 66 i -65 1-64 1-62 i'59 '58 56 '53 1-50 4 6 '44 '41 56 179 179 177 176 75 174 172 I'70 1-68 66 6 4 61 i'58 '55 52 '49 CO 1-89 2'OO 1-89 2'00 1-87 1-98 1-86 1-97 85 96 1-84 1-94 1-82 1-80 1*92! 1-90 178 1-88 76 86 11 70 80 1-67 177 63 74 6 1 70 i 62 2-13 2*12 2'II 2IO 2-08 2-07 2-05 2-03 2-00 i '97 1-95 1-91 1-88 84 81 77 64 2-28 2-27 2'26 2-24 2-23 2-21 2-19 2-17 2-14 2-II 2-08 2-05 2'01 '97 '93 89 MINUTES OP HOUR-ANGLE. [REDUCTION. N. 5m 6m 7m 8m 9m 10m llm 12m 13m 14m 15m 16m 17m 18m 19m 20m 050 0-4 0*5 0-8 I'O % 3 1-6 2*0 *"3 27 3*2 37 4-2 47 5 '3 s;9 6'5 055 0-4 o'6 0-9 i'i '4 1*8 2 '3 2*5 3'5 4-0 4*6 5-2 7-2 0-60 0*5 07 I'O i*3 '6 2'0 2-4 2-8 3' 3 3'8 4 '4 5"o 57 6*4 7'' 7 '9 65 O*'* 07 I'O I '4 7 2*1 2-6 3*o 3'5 4'i 4*8 5*4 6-1 6'9 77 8-5 0-70 0-6 0-8 I'l '8 2-3 2'8 3'3 3'8 4'5 5*2 5'9 6'6 7 '4 8-3 9-2 0-75 0-6 0-8 I '2 r6 "9 3-0 3*5 4*1 4'8 5'5 7'i 7'9 8'8 0-80 0-6 0-9 1*3 17 2'I 2'6 3*2 3 '8 4*4 5-1 5 '9 6.7 7'6 8'5 9'4 10-4 0-85 07 0-9 2'2 2*8 3 '4 4*0 4-6 5*4 6-2 8-0 9-0 I0'0 I I'O 0-90 07 I'O i '4 i -9 2'3 3*1 3-6 4-2 4'9 6-6 7'5 8 '5 9 '5 io'6 11*7 0-95 07 I'O i'S 2'0 2*4 3 '2 3'8 4*4 5-2 6-1 7-0 7;9 9-0 10 '0 II'2 12-3 1-00 0-8 I 1-6 2*1 2'6 3*3 47 5 '5 6-4 7*4 9*5 io'6 i I'S 13 'o 1-05 0-8 I 17 2'2 27 3 '4 4*2 4 '9 57 68 77 8'8 9-9 in 12-4 i3'6 1-10 0.9 '2 1-8 2'3 2-8 3 '6 4'4 5' 1 6-0 7-0 8-1 9-2 10-4 117 13*0 i4'3 1-15 I'O '3 i*9 3-0 3*8 5]3 6-3 7*4 8-5 9-6 10*9 I2'2 13-6 15-0 1.20 I'O '3 2-5 4-0 4-8 6-6 77 8-9 1 1 '4 127 14-2 '57 1-23 I'O i'4 2*0 2-6 3-3 4'i 5-0 5-8 6-8 8-0 9-2 10-5 u-8 I3'2 147 16-3 For Lat. or Alt. above GO see Table (iii.), p. 35 33 TABLE III. EX-MERIDIAN TABLE LATITUDE. Second Correction o o o o o 42 o o o o o o Reduction Alt. 34 36 38 40 44 46 48 50 52 54 56 58 60 Dec / / , 5 10 15 Q N N N N N N N N N N N N N N 8 0-84 0'82 0'80 077 075 073 071 0-68 0'6 5 0-62 0-56 '53 0-50 4 i O'O 0.0 i O'O 10 0'84 0'82 0-80 0'77 075 073 071 0-68 0'6 5 O-62 O'5c 0- 5 6 '53 0-50 8 O'O o-i O'l 12 0'85 0'82 0-80 078 076 073 071 0-68 0-66 0-6l o-oc o'54 0-5I 10 O'l 0'2 O'2 4 0-86 0-83 0-81 079 077 074 072 0.69 0-66 0-63 o-6c '54 0-5I 12 O'l 0'2 O'3 16 0-86 0-84 0-82 0'80 078 075 073 070 0-67 O'6<: 0*61 0'58 '55 0-52 14 O'l C'3 J '4 18 0-87 0'85 0-82 0'80 078 075 073 0.70 0-67 0-64 0-61 0- 5 8 0'55 0-52 16 O'2 0-4 0-6 20 22 0'88 0-89 0-86 0-87 0-84 0-85 0-8 1 0-82 079 0-80 7 6 077 074 075 071 072 0-68 0-69 0-65 0-66 0-62 0-63 0-59 o'6o 0-56 0-57 o : 53 18 20 O'l 0.5 0.6 07 0-9 2 4 0-90 0-88 0-86 0-83 0-8 1 7 8 076 073 070 0-67 0-64 0-61 0-58 o'54 22 0-4 0.7 n 26 28 092 0-94 0-89 0-91 0-87 0-89 0-85 0-87 0-82 0-84 079 0-81 077 079 074 076 071 073 o-6S 070 0-65 0-67 0-62 0-63 0'59 o 60 o-55 0-56 24 26 o-'s 0-9 ro *'3 1-5 30 0-96 _vr>C 0-93 _.._/: 0-91 0-88 0-86 _.oo 0-83 _.o r 0'80 -.0- 0.77 074 -.. /' 071 0-68 0-64 _v* o - 6i _ ,/r _ o-57 28 0-6 32 34 o 90 I'OO 096 0'98 0-93 o'9S 0-90 0*92 O oo 0-90 085 0-87 O 82 0'8 4 079 o - 8i o 76 078 073 074 0*70 071 O'OO 0-67 0-63 0-64 o*59 0-60 Reduction 36 I '03 I'OO 098 0-95 0'Q2 0-89 0'87 0-83 0-80 076 '72 0-69 0-65 0-62 O J * ^j Deo. 38 I -06 03 oo 0-97 0'94 0-91 0-88 0-85 0-82 078 075 071 0-67 0.63 20 25 30 40 1-09 06 03 oo O 97 0'94 0-91 0-88 0-85 0-81 077 073 0-69 065 42 09 06 03 1*00 0-97 0'94 0-90 0-87 0-83 079 075 071 0-67 o f t j 4 O'O O'l O'l 44 1-15 '12 09 06 1-03 I -00 0-97 0-93 0-90 0-86 0-82 078 074 0-69 8 0'2 0-2 0-3 46 1-19 16 '*3 10 1-07 1-03 oo 0-96 0-93 0-89 0-85 0-80 076 071 10 O*3 O'A *5 48 1-24 21 18 14 I ii 1-07 03 I'OO 0-97 0-92 0-88 0-83 079 074 12 0*4 'S 07 50 1-29 26 23 19 I 15 I'll 08 04 roo 0.96 0-92 0-87 0-82 077 14 0'6 07 0-9 52 1-34 '3 1 28 24 I 20 1-16 12 08 1-04 1. 00 0-96 0-91 0-86 c-8i 16 0-8 ro I'2 54 1-41 38 34 29 I 26 1-22 18 14 no 1.05 roo 0-95 0-90 0-85 18 ro I'2 JC 56 1-49 1-45 41 i'37 I 33 1-29 25 20 1-15 no 05 oo 0-95 0-89 20 I'2 1-5 r8 ''57 *'53 49 i '45 1-41 1-36 32 27 i n6 ii 06 I'OO 0-94 22 le r8 2'2 60 i '66 i '62 58 i'53 I 49 I -44 '39 *34 1-29 1-23 18 12 I -06 I'OO 24 17 2'2 2'6 62 177 '73 1*69 1.64 I 59 1-54 48 i" 8 3 1-31 26 19 1-13 1-07 26 2'O 2-5 3-0 64 1-89 1-84 i -So 1-75 1 69 1-64 58 1-3 3 1-47 1-40 '34 27 I-2I 1-14 28 2'3 2-0 3'5 MINUTES OF HOUR- ANGLE DEDUCTION N. 5m 6m 7m 8m 9m 10 m llm 12m 13m L4m; L5m 16m 17m 18m 19m 20m 1-23 '0 *4 2'0 2'6 1 i 3'3 4'i 5 'o 5 ( 8 6-8 8 '-o 1 9'2 10-5 ii-8 13-2 147 16-3 1-30 o '4 2'I 27 3'4 4'3 5*2 6-1 7-1 8-3 9-6 12-3 13-8 17-0 1-33 o '4 2*1 2'8 3'5 4'4 5'4 6'3 7 '4 8-6 ] O'O 11-3 12-8 I4-3 IS'9 17*6 1-40 I '5 2 '2 2-9 3'6 4-6 5-6 6-6 77 9-0 ] 0-4 11-9 *3'3 14-8 16-5 18-3 1-43 'I 5 2'3 3*0 37 47 5 '8 6'8 7'9 9'3 i 07 I2'2 137 *5'3 17-1 18-9 1-50 '2 6 2'4 3'i 3'9 4*9 6-0 7-0 8'2 9-6 i i'i I2'C 14-2 I5-9 177 19-6 1-53 '2 7 2 '5 3' 2 4-0 5-1 6'2 7 -2 8-5 9-9 i i '4 I3'0 147 16-4 18-3 20-3 1-60 2 8 2'6 3 '4 4*2 5-3 6' 4 7'5 8-8 O'2 1 i '8 15-2 17-0 18-9 21'0 1-63 '3 8 2'6 3'5 4'3 S'4 6-6 77 9-0 0-5 i 2'2 I3'8 15-6 17-5 i9'5 21-6 1'70 "4 i '9 27 3-5 4*4 5*6 6-8 8-0 9'3 0-9 2'6 I4'3 16-1 180 20'I 22 '2 1-73 '4 i -9 37 4*5 57 7'0 8-2 9-6 I'2 1 147 16-5 *S 2O'6 22'8 1-80 *4 2'0 2-9 47 5'9 7'2 *5 9-9 i'5 i 3-3 I5-I 17-0 19-1 21'2 23-8 1-83 '4 2*0 2 -9 3 '9 4'8 6-1 7'4 87 IOT rS 37 I5'5 17-5 19-6 21-8 24-I 1-90 '5 2'I 3'o 4-0 4'9 6-3 8-9 10-4 2'I 4' 1 i6'O 18-0 2O'I 22*4 24'8 1'93 2*1 4*1 5-0 6-4 '7-8 9-1 107 2'4 4'4 16-4 18*5 2O'6 2 3 '0 25^ 2-00 i-6 2-2 3*2 4-2 5-2 6-6 8-0 9 '4 iro 2'8 4'8 16-8 19-0 21'2 23-6 26 -a ( For Lat. or Alt. above 60 see Table (iii.), p. 35 34 TABLE III. EX-MERIDIAN (CONTD.; MINUTES OF HOUR-ANGLE M M M M M M M M M M M M M M M M 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 / t . . . i f t . i 050 6-5 7-2 7.9 8-6 9'5 10-2 ii'i 12-0 12-9 i3;8 147 157 16-8 17-8 18-9 20-1 055 7-2 8-0 87 9*5 10-4 1 1"3 I2'2 13*2 14-1 16-2 I7-3 18-4 19-6 20-8 22'2 060 7-9 87 9'5 10-4 ii'3 12-3 i3'3 I4'4 I5-4 16-5 177 l8'9 20-2 21-4 22-8 24-2 065 8-5 9 '4 10-3 1 1 '2 I2'2 I3-3 14-4 I 5 '6 167 17-9 19-1 20'4 21-8 23-2 24-6 26-2 070 9-2 lO'I ii-i I2"I 13*2 H'3 16-8 17-9 19-3 20-6 22-0 23-6 25-0 26-6 28-2 075 9-8 10-9 11-9 12-9 I4-I I5-4 16-6 18-0 19-2 20-6 22-1 23-6 25-2 26-8 28-4 30-2 080 10-4 11-6 127 13-8 I5'I 16-4 17-8 19-2 20-5 22-0 23-6 25-2 26-8 28-6 30-4 3 2-2 085 iro 12-3 I 3'5 14-6 16-0 17-4 18-9 20-4 21'8 23-4 25-0 26-7 28-4 30-2 3 2-2 34' 2 090 117 13-0 14-3 I5-5 17-0 18-4 2O'O 21-6 23-1 24-8 26-5 28-3 30-2 32-0 34*o 36-2 095 12-3 13-8 I5 -I 16-4 17-9 I9'5 2I-I 22-8 24-4 26-2 28-0 29-9 31-8 33-8 36-0 38-2 100 13-0 14-4 15-9 17-3 18-9 20-5 22-2 24-0 257 27-6 29-5 33-6 35 ' 6 38-0 40-2 105 13-6 15-2 16-7 18-1 19-8 21-5 23*3 25-2 27-0 28-9 30-9 33'0 35'4 37'4 39'8 42-2 1 10 14-3 15-9 I7-5 19-0 20'8 22'5 2 4 '4 26-4 28-3 30-3 32-4 347 37-0 39'2 41-8 44*2 1 15 120 15-0 157 16-7 I7-3 19-1 19-8 20-8 217 22-7 23-6 24-6 III 27-6 28-8 29-6 30-9 317 33'9 35'4 36-2 37'9 38-6 40-4 41-0 42-8 43'6 46-2 48-2 125 16-3 18-1 19-9 21-6 23-6 25-6 277 30-0 32-2 34-4 36-8 39'5 42-0 44-6 47'4 50-2 130 17-0 18-8 20-7 22-5 24-6 26-6 28-9 31-2 33-5 35-8 38-3 40-9 43-6 46-2 49'4 52-2 135 17-6 19-6 21-5 23-3 25-5 277 30-0 32-4 347 37'2 39-8 42-5 45' 2 48-0 51-2 54'2 140 18-3 20-3 22-3 24-2 26-5 287 3I-I 33-6 36-0 38-6 4i-3 44-1 47-0 49-8 53'2 56-2 1-45 18-9 20-9 23-1 25-0 2 7 -4 297 32-2 34-8 37'3 40-0 42-7 48-6 51-4 58-2 150 19-6 21-7 23-9 25-9 28-4 307 33'3 36-0 38-6 41-4 44-2 47'2 50-2 53'4 57'o 60-2 1-55 160 20-3 21-0 22-5 23-2 247 25-5 26-8 277 29-3 3O-2 31-8 32-8 34'4 35'5 38-4 39-8 41-1 427 44-2 457 47-2 48-8 50-4 52-0 53-8 57-0 58-8 60-8 62-3 64-4 1-65 21-6 23-9 26-3 28-5 3I'I 33-8 36-6 39*6 42-4 45'5 48-6 51-9 55'4 58-8 62-6 66-4 170 22'2 247 27-1 29-4 32-1 34-8 37'8 40-8 437 46-9 50- 1 53'5 57'2 60-8 64-6 68-4 175 22-S 27-9 30-2 35;8 38-9 42-0 4.S'0 48-2 58-8 62-4 66-4 70-4 180 23'4 26-1 28-7 31-1 34'0 40-0 43'2 49'5 5P i 567 60-4 64-0 68-4 72-4 1-85 24-1 26-8 29-5 3i'9 34'9 37'9 41-1 44'4 47'5 50-9 54'5 58-2 62-0 65-8 70-2 74'4 190 24-8 27-5 30-3 32-8 35'9 ; 38-9 42-2 45-6 48-8 56-0 59'8 63-8 67-6 72-2 76-4 SECOND CORRECTION [EEDUCTION DEC 35 40 45 50 55 60 62 64 66 68 70 72 74 76 78 80 4 o-i o-i o-i o-i C'l O-I 0-2 0-2 0-2 0-2 0-2 O'2 O'2 0-2 0-2 O'2 8 0-3 0-4 0-4 0-5 0-5 o'6 0-6 0-6 0-6 0-7 0-7 0'7 0'7 0-7 0-8 0-8 10 0-6 0-7 0-8 0-8 0-9 0-9 i-o I'O I'O I-I I-I I'l [-2 1-2 1-2 12 4 M 0-8 0-9 ro i-i 1-2 1-3 1*4 w -O i '4 i-4 i'5 1-5 1-6 1-6 17 1-7 17 14 I'U I 2 I 3 I C i"6 i"8 I 1-9 2'O 2'O 21 21 2'2 ^3 ^ *? 2 4 16 i '4 1*5 17 1-9 2-1 2> 3 2-4 2'5 2-6 2-6 2-7 2-8 2'9 9 | 3 3' 1 18 1-7 2-0 2'2 2-4 2'7 2-9 3-0 3-1 3'2 3'3 3'4 3'5 3-6 37 3'8 3'9 20 2"! 2'4 2'7 3-0 3'3 3-6 37 3-9 4-0 4-1 | 4-2 4-3 4-5 4-6 47 4-8 22 2'5 2-9 3'3 36 4-0 4'4 4'5 47 4-8 5'0 5' 1 5-2 5'4 5'5 i 57 5'S 24 3 -0 3-5 3'9 4'3 4-8 5' 2 5'4 5'5 57 5'9 6-1 6-2 6-4 6-6 6-7 6-9 26 3'5 4-0 4-6 5*6 6-1 6-7 6-9 7*i 7-3 7'5 77 I 7'9 8-1 28 47 5*3 5'9 6-4 7-0 7'3 7'5 77 8-0 8-2 8-4 87 8-9 9-1 9'4 NOTE. Should the Hour~A.ngle exceed 35m. take out the correction for its half and multiply it by 4. 35 SUPPLEMENTARY TABLES. M EXTENSION OF TABLE I. TO LAT. OR DEC. 80. For Lat. o 60 62 63 o 6 4 o 65 o 66 o 67 o 68 o 6 9 o 70 71 o 72 o 73 o 74 o 75 o 76 o 77 78 o 79 o 80 TakeL.it. o 41 F 43 O 44 o 45 o 47 o 48 50 51 o 52 o 54 55 o 57 o 58 o 49 o 51 o 53 o 56 o 58 o 46 o 48 And Mult. Nr. by 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 5 5 (ii.) EXTENSION OF TABLE II. TO LAT. 80. For Lat. & o 62 o 63 o 6 4 o 65 66 6 7 68 o 6 9 o 70 71 72 o 73 o 74 o 75 76 o 77 78 o 79 80 Take Lat. o 14 o 20 25 o 29 o 32 35 2 o 38 41 o 44 o 47 o 50 o 52 o 54 2 o 56 o 39 o 44 47 3 O 50 3 o 55 o 59 3 And Mult. Corr. by L 2 2 2 2 2 2 2 2 2 2 3 3 3 (iii.) EXTENSION OF EX-MERIDIAN TABLE TO LAT. 80 OR ALT. 80. For Lat. o 61 o 62 o 63 64 o 65 o 66 67 o 68 o 6 9 o 70 71 72 73 74 o 75 o 76 o 77 78 79 80 Take Lat. ,8 o 22 o 27 O 30 o 33 o 36 40 o 43 o 46 48 c 50 o 52 54 o 56 58 45 4S o 52 56 o 60 And Div. Redn. by For Alt. il O 61 2 o 62 2 o 63 2 o 6 4 2 o 65 2 66 2 ~b 2 68 2 o 6 9 2 o 70 2 71 2 o 72 2 o 73 2. o 74 2 O 75 3 3 77 3 78 3 79 3 80 Take Alt. o 14 o 20 2 25 29 32 o 35 38 O 42 2 o 44 2 47 2 5 2 52 2 54 2 56 2 59 2 o 43 o 47 o 53 o 55 o 58 3 And Mult. Redn. by b 2 2 2 2 2 3 3 3 3 FOR COMBINED EX-MERIDIANS. (iv.) LESSER HOUR-ANGLE Inter- val. M 5 M 6 M 7 M 8 M 9 M 10 M 11 M 12 M 13 M 14 M 15 M 16 M 1 M 17 1 18 M 19 M 20 M 10 '5 o'6o 070 0-80 0-90 I'OO no I'20 1-30 i '40 i'So i -60 170 80 90 2'OO 12 0-42 '5 0-58 0-66 075 0-83 0-91 0-99 i -08 n6 1-24 1-33 1-41 '49 S r66 14 0-36 o'43 0-50 0-58 0*65 072 079 0-85 0-93 I'OI i -08 'I5 1*21 28 36 i'43 16 0-31 o'37 o'43 0-49 0'K 0*62 0-68 074 0-80 0-87 0-94 I'OO I -06 ii 19 1-25 18 0-28 o'34 0-39 0'4,S 0-50 0-56 0-62 0-67 073 078 0-84 0-89 0'95 oi 06 l'I2 2O 0-25 0-30 o'35 0-40 0'45 0*50 o'55 0-60 0-65 070 075 0-80 0-85 0-90 0'95 TOO 22 O'22 0-26 0-31 0'35 0-39 0-44 0-48 Q'S3 o'S7 0*61 0-66 070 075 079 0-83 0-88 24 O'2O 0-24 0-28 0-33 0-37 0-41 0-45 0-49 o-S3 o-57 O'6i 0*65 o't>9 073 077 081 26 O-I9 0-22 0-26 0-30 0'34 Q'37 0-41 0'45 0-49 0-52 0-56 o'6o 0-65 0*69 073 077 28 O-i; 0-21 0-24 0-27 0-31 o'34 0-38 0-41 0-44 0-48 o-si o-ss 0-58 0-62 0-65 o-6S 3O 0-16 0'20 0-23 0-27 0-30 o\33 o'37 0-40 o'43 0-47 0-50 0> 53 o'57 o - 6o 0*63 0-67 NOTE. Should either the Hour-Angle or interval exceed the limit of the table, enter it with the half or third of both. Thus for H. A. 18m. and interval 48m. we have 9m. and 24m. which give -37. 36 ABBEEVIATED TABLES. (IV.) LOG-SECANTS. PARTS FOR MINUTES. i Q 6 I'D 20 30 40 50 1 10 20 30 40 50 i 2 '3 4' 5 6 7 8 9 o O'OOOO oooo OOOO 0000 0000 oooo 1 O'OOO I r xx>i 0001 0001 0001 0002 O o o I i 2 0-0003 0003 0004 0004 0005 0005 3 0-0006 3007 0007 OOO8 ; OOO9 OOIO i i i 4 O'OOI I OOII 0012 0013 0014 0015 5 0-0017 0018 0019 OO2O ! OO2 I 0023 o o o I i 1 6 8 0-0024 0-0042 0025 0044 0027 0046 0028 0048 0029 0050 0031 0052 7 9 0-0032 0-0054 3034 0056 0036 0058 0037 0039 0060 0062 0041 0064 o o o I I I J i i i i 2 2 10 0-0066 0069 0071 0073 0076 0078 11 0-0081 0083 0086 0089 0091 0093 o o I 2 2 2 2 12 0-0096 0099 OIOI 0104 0107 OIIO 13 0-0113 0116 0119 OI22|OI25 0128 I 2 2 2 3 14 0-0131 0134 0137 0141 0144 0147 15 0-0151 0154)0157 0161 0164 0168 2 2 2 3 3 16 0-0172 0175 0179 0183 0186 0190 17 O-OI94 0198 O2O2 0206 02 i o 0214 o 2 2 2 3 3 18 0-0218 O222 0226 0230 0235 0239 19 0-0243 0248 0252 0257 0261 0266 2 2 2 3 3 3 20 0-0270 0275 0279 0284 0289 0294 21 0-0298 0303 0308 0313 0318 0323 I 2 2 3 3 4 4 22 0-0328 0334 0339 0344 0349 0354 23 0-0360 0365 0371 0376 0382 0387 2 2 3 3 4 4 5 24 0-0393 0398 0404 0410 0416 0421 25 0-0427 0433 0439 0445 0451 0457 2 2 3 3 4 5 5 26 0-0463 0470 0476 0482 0488 0495 27 O-O5OI 0508 0514 052110527 0534 2 3 3 4 4 5 6 28 0-0541 0547 0554 0561 0568 0575 29 0-0582 0589 0596 0603 0610 0617 2 3 3 4 5 6 6 30 0-0625 0632 0639 0647 0654 0662 31 0-0669 0677 0685 0692 0700 0708 2 2 3 4 5 5 6 7 32 0-0716 0724 0732 0740 0748 0756 33 0-0764 0772 0781 0789 0797 0806 2 2 3 4 5 6 6 7 34 0-0814 0823 0831 0840 0849 0858 35 0-0866 0875 0884 0893 0902 0911 2 3 3 4 5 6 7 8 36 0-0920 0930 0939 0948 0958 0967 37 0-0976 0986 0996 1005 1015 IO25 2 3 4 5 6 7 8 9 38 0-1035 1045 1055 1065 1075 1085 39 0-1095 1105 1116 1126 1136 1147 2 3 4 5 6 7 8 9 40 0-1157 1168 1179 1190 1200 I2II 41 0-1222 1233 1244 1255 1267 1278 2 3 4 5 7 8 9 10 42 0-1289 1301 1312 1324 1335 1347 43 0-1359 1382 1394 1406 1418 2 4 5 6 7 8 9 ii 44 0-1431 1443 1455 1468 1480 1493 45 0-I505 1518 I53 1 1543 1556 1569 2 4 5 6 7 9 10 ii 46 0-! 5 82 1595 1609 1622 1635 1649 47 0-I662 1676 1689 1703 1717 1731 3 4 5 7 8 9 ii 12 48 0-1745 1759 1773 1787 1802 1816 49 0-I83I 1845 1860 1875 1889 1904 2 3 5 6 7 9 10 12 13 50 0-1919 1934 1950 1965 1980 1996 51 0-201 1 2027 2043 2058 2074 2O9O 2 3 5 6 8 9 ii 12 14 52 0-2107 2123 2139 2156 2172 2189 53 C-2205 2222 2239 2256 2273 2290 2 3 5 7 8 10 12 13 15 54 0-2308 2325 2343 2360 2378 2396 55 0-2414 2432 2450 2469 2487 2506 2 4 5 7 9 ii 13 14 16 56 0-2524 2543 2562 2581 26OO 2620 57 0-2639 2658 2678 2698 2718 2738 2 4 6 8 10 12 14 16 18 58 0-2758 2778 2799 2819 2840 2861 59 0-2882 2903 2924 2945 2967 2988 2 4 6 8 10 13 15 17 19 60 0-3010 3032 3054 3077 3099 3122 61 0-3I44 3167 3190 3213 3237 3260 2 5 7 9 ii M 16 18 21 62 0-3284 3308 3332 3356 3380 3405 63 0-3429 3454 3479 3505 3530 3556 2 5 7 10 12 15 17 2C 22 64 ,-3582 3608 3634 3660 3687 37H 65 0-3741 3768 3795 3823 3851 3879 3 5 8 1 1 13 16 19 22 24 66 0-3907 3935 3964 3993 4O22 4052 67 0-4081 4111 4141 4172 4202 4233 3 6 Q 12 15 18 21 24 27 68 0-4264 429614327 4359 4391 4424 69 0-4457 4490 4523 4557 4591 4625 3 7 10 13 16 20 2 3 26 3 6 5 10 15 20 25 i 2 3 i i 30 35 40 45 50 55 111 4 70 0-4659 4677 4694 4712 4730 4747 4 7 ii 14 0-476! ,4783 4801 4819 483; U855 4 r / II 15 71 0-4874 4892 4910 4929 4948 4966 4 7 ii 15 0-498. ; 5004 5023 5042 5061 5081 4 * > II 15 72 0-5100 5120 5139 5159 5179 5199 4 8 12 16 0-5210 ) 5239 5259 5279 5300 5320 4 j \ 12 16 73 0-5341 5361 5382 5403 5424 5445 t 8 12 16 0-546- ^5488 5509 5531 555315575 4 c > 13 17 74 0-5597 5619 5641 5663 5686 57o8 c 9 13 18 o-573 [ 5754 5777 5800 582. J 5847 5 | ) 14 1 9 75 0-5870 5894 5917 594i 5965 599C 5 10 M 19 16038 6063 6088 611 36138 5 1C ) 15 20 76 06163 6189 6214 6240 6266 6292 5 10 16 21 0-631} '6345 6371 6398 642 56452 5 I] 16 22 77 0-6479 6507 6534 6562 659C 6618 6 ii 17 22 0-664' 76675 6704 6733 676 26792 6 11 5 17 2 3 78 0-6821 6851 6881 6911 6942 6973 6 12 24 0-700. 37035 7066 7098 7i3< 3 7162 6 i; ; 19 25 79 0-7194 7227 7260 7293 7326 735* \ 13 2C 26 0-739, 17428 7462 7497 753 27568 7 \L I 21 28 80 0-7603 7639 7676 7712 774< 7786 * 15 22 30 0-782, 17862 7900 7939 797 3 8017 8 \\ ; 23 31 81 0-8057 8097 8137 8178 82IC > 8261 8 16 24 33 0-830 38345 8388 8432 847 58520 9 l\ r 26 35 82 0-8564 8610 8655 8701 874* > 879= 9 18 2 37 0-884. 58891 8940 8985 9039; 9090 10 2C ) 30 39 83 0-9141 9'93 9245 9298 935- . 940^ 1 1 21 32 42 0-946 f 95*7 9574 9631 96899748 ii 23! 34 46 84 0-9808 9868 9930 9992 1-005" ) OIK 12 25 37 50 102 5 C 0318 0386 045 50525 14 2; J 41 55 85 0597 0670 0744 0819 0896 i 097^ 15 30 4 C 60 1-105- \ "3* 1217 1301 138 7 1475 17 3< \ 50 67 86 -'564 1655 1749 1844 1941 2041 1 9 38 57 76 1-214 3 2245 2355 2465 257 7 2693 22 4 1 66 88 87 2812 2934 3060 3190 332; 5 346] 26 52 7* ; 104 1-360 3375C 3903 4061 42244395 32 6 3 95 126 88 4572 4757 4950 5152 536: J 558< 4i 81 12: 162 1-582 i 6o6c 6332 6612 6912 7234 57 1131169 226 89 7581! 7959 8373 8831 9342 9922 92 183 27 5396 2-059 2 138; 2 352 3602 5363 8373 2SC 500 750 [000 37 FOR LONGITUDE BY CHRONOMETER.- (V.) HALF LOG-HAVERSINES. o Q 6 5 10 15 20 25 i 2 3 i 30 35 40 45 50 55 i i 3 4 O'OOO 1-862 2*163 2-339 2-464 2*561 35 70 106 141 2-640 2-707 2-765 2-816 2-862 2-903 10 20 3i 41 1 2-941 2-976 3-008 3-038 3-066 3-092 6 12 18 24 1 3"i 17 3-140 3-163 3-184 3-204 3-223 4 8 M 17 2 3-242 3-260 3-277 3-309 3-324 3 6 10 13 2 3-339 3-353 3-367 3-393 3-406 3 5 g 10 3 3-418 3-430 3-441 3-453 3-464 2 4 7 9 3 3-495 3-505 3*5^5 3-524 3-534 2 4 6 8 4 3-543 3-552 3-561 3-569 3-578 3-586 2 3 5 7 4 3-594 3-602 3-610 3-617 3-625 3-632 I 3 4 5 5 3-6397 6468 6539 6608 6677 6744 H 28 42 55 5 3-6810 6876 6940 7003 7066 7127 13 25 38 5I 6 3-7188 7248 7307 7365 7422 7475 12 23 35 46 6 3-7535 759i 7645 7699 7752 7805 II 22 33 43 7 37857 7908 7959 8009 8059 810* 10 20 30 40 7 3-8156 8204 8251 8298 8345 8390 9 19 28 37 8 8481 8525 8569 8613 8656 9 18 27 35 8 3-8699 8741 8783 8824 8865 8906 8 16 25 33 9 3-8946 8986 9026 9065 9104 9H3 8 16 24 9 9219 9256 9293 9330 9367 7 15 22 29 10 3-9402 9439 9475 9510 9545 9580 7 14 21 28 10 3-96i4 9649 9682 9716 975 9783 7 14 20 27 11 3-9816 9848 9881 9913 9945 9977 6 13 19 26 11 4-0008 0039 0070 OIOI 0132 0162 6 12 IS 2i 12 4-0192 0222 0252 0282 0311 0340 6 12 18 24 12 4-0369 0398 0426 0455 0483 0511 6 12 17 23 13 4-0539 0566 0594 0621 0648 0675 5 II 16 22 13 4-0702 0728 0755 0781 0807 0833 5 10 16 21 14 4-0859 0885 0910 0935 0961 0986 5 IO 15 2O 14 4-1011 1035 1059 1084 1109 "33 5 IO 15 2O 15 4-1157 IlSl 1205 1228 1252 1275 5 9 14 18 15 4-1299 1322 1345 1368 1390 1413 5 9 14 _ I 16 4-1436 1458 1480 '502 | 1525 1546 4 9 13 18 16 4-1568 1590 1612 1633 1676 4 9 13 17 17 4-1697 I7l8 1739 1760 1781 1801 4 8 13 17 17 4-1822 1842 1863 1883 190^ 1923 4 8 12 16 18 4-1943 1963 1983 2003 2O2 2 2042 4 8 ii 15 18 4-206 1 2081 2IOO 2119 2138 2157 4 8 II 15 19 4-2176 2195 2214 2232 2251 2269 4 8 ii 14 19 4-2288 2306 2324 2343 2361 2379 4 7 II 14 o w ; 1 , . 1 . 3 . . / i ( t * t i . i \ J / 10 20 30 40 50 Q 10 20 30 to 50 1 2 3 4 5 6 1 8 9 o 20 4-2397 2432 2468 2503 2538 2572 21 4-2606 2640 2673 2707 2740 2773 3 7 IO 14 17 21 24 27 31 22 4-2806 2838 2870 2902 2934 2965 23 4-2997 3027 3058 3089 3"9 3M9 3 6 9 12 16 19 21 25 28 24 4-3179 3208 3238 3267 3296 3325 25 4-3353 3382 3438 3466 3493 3 6 9 II 14 17 20 23 26 26 4-3521 3548 3575 3602 3629 3655 27 4-3682 3708 3734 376o 3786 3811 3 5 8 II 13 16 19 21 24 28 4-3837 3862 3887 3912 3937 396i 29 4-3986 4010 4034 4059 4083 4106 2 5 7 IO 12 15 17 20 22 30 4-4130 4153 4177 4200 4223 4246 31 4-4269 4292 43M 4337 4359 438i 2 5 7 9 II 14 16 18 21 32 4-4403 4425 4447 4469 4490 4512 33 4-4533 4555 4576 4597 4618 4639 2 4 6 9 II i ' I t; 17 19 34 4-4659 4680 4700 472i 476i 35 4-4781 4801 4821 4841 4861 4880 2 4 6 8 10 12 14 16 18 36 4-4900 4919 4939 4958 4977 4996 37 4-5015 5034 5052 5071 5090 5108 2 4 6 8 9 II 13 15 17 38 4-5126 5145 5163 5181 5199 5217 39 5253 5270 5288 5306 5323 2 4 5 7 9 II 13 14 16 40 4-5341 5358 5375 5392 5409 5426 41 4-5443 546o 5477 5494 55io 5527 2 3 5 7 8 IO 12 14 15 42 4-5543 5560 5576 5592 5609 5625 43 4-5641 5657 5673 5689 5704 5720 2 3 5 6 8 10 II 13 14 44 4-5736 5751 5767 5782 5798 45 4-5828 5844 5859 5874 5889 5904 2 3 5 6 8 9 II 12 14 46 4-5919 5934 5948 5963 5978 5992 47 4-6007 6021 6036 6050 6065 6079 3 4 6 7 9 10 12 13 46 4-6093 6107 6121 6i35 6149 6163 49 4-6I77 6191 6205 6219 6232 6246 3 4 6 7 8 IO II 13 50 4-6259 6273 6286 6300 6313 6327 51 4-6340 6353 6366 6379 6392 6405 3 4 5 7 8 9 10 12 52 4-6418 6431 6444 6457 6470 6483 53 4-6495 6508 6521 6533 6546 6558 3 4 5 6 8 9 10 12 54 4-6570 6583 6595 6607 6620 6632 55 4-6644 6656 6668 6680 6692 6704 2 4 5 6 7 8 IO II 56 4-6716 6728 6740 6752 6763 6775 57 4-6787 6798 6810 6821 6833 6844 2 3 5 6 7 8 9 10 58 4-6856 6867 6878 6890 6901 6912 59 4-6923 6935 6946 6957 6968 6979 2 3 4 6 7 8 9 IO 60 4-6990 7001 7012 7022 7033 7044 61 4-7055 7065 7076 7087 7097 7108 2 3 4 5 6 7 8 IO 62 4-7118 7129 7139 715 7160 7171 63 4-7181 7191 7201 7212 7222 7232 2 3 4 5 6 7 8 9 64 4-7242 7252 7262 7272 7282 7292 65 4-7302 7312 7322 7332 7342 7351 2 3 4 5 6 7 8 9 66 47361 7371 738o 7390 74oo 7409 67 4-74I9 7428 7438 7447 7457 7466 2 3 4 5 6 7 8 9 68 4-7476 7485 7494 7504 7513 7522 69 4-7531 7540 7550 7559 7568 7577 2 3 4 5 5 6 7 8 70 4-7586 7595 7604 7613 7622 7631 71 4-7640 7648 7657 7666 7675 7683 2 3 4 4 5 6 7 8 72 4-7692 7701 7710 7718 7727 7735 73 4'7744 7752 7761 7769 7778 7786 2 3 3 4 5 6 7 8 74 4-7795 7803 7811 7820 7828 7836 75 7853 7861 7869 7877 7885 2 2 3 4 5 6 6 7 76 4-7893 7901 7910 7918 7926 7934 77 4-7942 7949 7957 7965 7973 7981 I 2 2 3 4 5 6 6 7 78 7997 8004 8012 8020 8027 79 4-8035 8043 8050 8058 8066 8073 I 2 3 4 5 5 6 7 * Vide Examples page 43, worked by INMAN'S, NOKIE'S, and THE SECANT METHOD. 38 ABBREVIATED TABLES. (V.) HALF LOG-HAVERSINES. u , 1 j C3 1 t 1 M Q 10 20 30 40 50 Kd Q 10 20 30 40 50 i 2 3 4 5 6 7 8 9 80 4-8081 8088 8096 8103 8111 8118 81 4-8125 8i33 8140 8148 8i55 8162 I 2 3 4 4 5 6 7 82 4-8169 8177 8184 8191 8198 8205 83 4-8213 8220 8227 8234 8241 8248 2 3 4 5 5 67 84 4-8255 8262 8269 8276 8283 8290 85 4-8297 8304 8311 8317 8324 8331 2 3 3 4 5 66 86 8345 8351 8358 8365 837J 87 4-8378 8385 8391 8398 8405 8411 2 3 3 4 5 5;6 88 4-8418 8424 8431 8437 8444 845 89 4-8457 8463 8469 8476 8482 8489 2 3 3 4 5 5 6 90 4-8495 8501 8507 85H 8520 8526 91 4-8532 8539 8545 855i 8557 8563 2 2 3 4 4 5 6 92 4-8569 8575 8581 8588 8594 8600 93 4-8606 8612 8618 8624 8629 8635 2 2 3 4 4 5 5 94 4-8641 8647 8653 8659 8665 8671 95 4-8676 8682 8688 869418699! 8705 2 2 3 3 4 5 5 96 4-8711 8716 8722 8728 8733 8739 97 4-8745 8750 8756 8761 8767 8772 2 2 3 3 4 4 5 98 4-8778 8783 8789 8794 8800 8805 99 4-8810 8816 8821 8827 8832 8837 2 2 3 3 4 45 100 4-8843 8848 8853 8858 8864 8869 101 4-8874 8879 8884 8890 8895 8900 2 2 3 3 4 4 5 102 4-8905 8910 8915 8920 8925 8930 103 4-8935 8940 8945 8950 8955 8960 2 2 3 3 4 4 104 4-8965 8970 8975 8g8o 8985 8990 105 4'8995 9000 9004 9009 9014 9019 1 2 2 3 3 4 4 106 4-9023 9028 9033 9038 9042 9047 107 4-9052 9056:9061 9066 9070 9075 o 2 2 3 3 4 4 108 4-9080 9084 9089 9093 9098 9102 109 4-9107 9111 9116 9120 9125 9129 o 2 2 3 3 4 4 110 4-9I34 9138 9142 9147 9i5i 9156 111 4-9160 9164 9169 9173 9177 9181 o 2 2 3 3 3 4 112 4-9186 9190 9194 9198 9203 9207 113 4-9211 9215 9219 9224 9228 9232 2 2 3 3 3 4 114 4-9236 9240 9244 9248 9252 9256 115 4-9260 9264 9268 9272 9276 9280 o 2 2 2 334 116 4-9284 9288 9292 9296 9300 9304 117 4-9308 93 1 2 9315 93'9 9323 9327 o 2 2 3 3 3 118 4-933I 9334 9338 9342 9346 9349 119 4-9353 9357 9364 9368 9372 2 2 3 3 3 120 4-9375 9379 9383 9386 9390 9393 121 4-9397 9401 9404 9408 94" 9415 o 2 2 333 122 4-9418 9422 9425 9429 9432 9436 123 4-9439 9442 9446 9449 9453 9456 2 2 2 3 3 124 4-9459 9463 9466 9469 9473 9476 125 4'9479 9483 9486 9489 9492 9496 2 2 233 126 4*9499 952 9505 9508 95 12 9515 127 9521 9524 9527 9530 9534 o 2 22 3 128 4-9537 9540 9543 9546 9549 9552 129 4-9555 9558 9561 9564 9567 9570 ol 2 2 2 3 130 4-9573 9576 9579 9582 9584 9587 131 4-9590 9593 9596 9599 9602 9604 2 2 2 3 132 4-9607 9610 9613 9616 9618 9621 133 4-9624 9627 9629 9632 9635 9638 2 2 2 3 134 4-9640 9643 9646 9648 9651 9654 135 4-9656 9659 9661 9664 9667 9669 2 2 2 3 136 4-9672 9674 9677 9679 9682 9684 137 4-9687 9689 9692 9694 9697 9699 2 2 2 138 4-9702 9704 9706 9709 9711 97H 139 4-9716 9721 9723 9725 9728 o 2 2 2 140 4-9730 9732 9734 9737 9739 974i 141 4-9743 9746 9748 975 9752 9755 2 2 142 4-9757 9759 9761 9763 9765 9767 143 4-9770 9772 9774 9776 9778 978o o 2 2 144 4-9782 9784 9786 9788 9790 9792 145 4-9794 9796 9798 9800 9802 9804 | 2 2 146 4-9806 9808 9810 9812 9814 9815 147 4-9817 9819 9821 9823 9825 9827 2i2 148 4-9828 9830 9832 9834 9836 9837 149 9841 9843 9844 9846 9848 o I 2 150 4-9849 9851 9853 9854 9856 9858 151 4-9859 9861 9863 9864 9866 9867 o 2 152 4-9869 9871 9872 9874 9875 9877 153 9880 9881 9883 9884 9886 I 154 9889 9890 9892 9893 9894 155 4-9896 9897 9899 9900 9901 9903 O I 156 4-9904 9905 9907 9908 9909 9911 157 4-9912 9913 9914 9916 9917 99i8 o O I I 158 4-9919 9921 9922 9923 9924 9925 159 4-9927 9928 9929 9930 9931 9932 o o I I 160 4-9934 9935 9936 9937 9938 9939 161 4-9940 9941 9942 9943 9944 9945 o o I ! 162 4-9946 9947 9948 9949 9950 163 9953 9954 9955 9956 9957 o C I 164 4-9958 9958 9959 9960 9961 9962 165 4-9963 9964 9964 9965 9966 9967 o o o O I 166 4-9968 9968 9969 9970 9971 9971 167 4-9972 9973 9973 9974 9975 9975 o c O I 168 4-9976 9977 9977 9978 9979 9979 169 4-9980 9981 998i 9982 9982 9983 o o o O I 170 4-9983 9984 9985 9985 9986 9986 171 4-9987 9987 9988 9988 9989 9989 o o I 172 4-9989 9990 9990 9991 9991 9991 173 4-9992 9992 9993 9993 9993 9994 o o o o o I 174 4-9994 9994 9995 9995 9995 9996 175 4-9996 9996 9996 9997 9997 9997 o o o o o o o o 176 178 4-9997 4-9999 9998 9999 9998 0990 9998 9999 9998 9998 9999 1 9999 177 179 4-9999 4-9999 9999 9999 9999 1 9999 9999 | 9999 9999 1 9999 9999 5.0000 o o o o o o 39 FOR LONGITUDE BY CHRONOMETER. (VI.) LOG HAVERSINES OF THE HOUR-ANGLE. P.M., OB WEST OF MERIDIAN. PABT8 FOB SECONDS. M M M M M M M M M M M S S s S s s s s a s | s s * 04 1 2 3 4 5 6 7 8 9 1 5 10 15 20 25 30 35 40 45 50 55 A.M. H M M H 00 O'OOO 4-678 C'28o ti'6l2 5 -882 6'O7^ 6*214 6-368 6'4846-<;86 50 10 6-678 T" W / ^ 6 1-760 J *-""' p ^3* 6-8^66'Qo; 2 "o^ 6"Q7O / 2 7 "O^O W *- J<-f 7-086 ** jw 7-n8 ** ^***fp* 0* JV/ 7*l87*23C 40 20 7-279 7-322 u ojuu yv-jj 7-3627-401 -// ^ 7-438 / ^^^ 7-473 7 '57 / * O 7'54o / ***^/ *5D 7-571(7-602 6 3 6 9 12 14 17 20 23 26 29 3i 30 30 7-631 7 '660 7-687 77i4 7-740 7765 7789 7-813 7-83617-859 '4 2 4 6 8 ii 13 15 17 19 21 23 20 40 7-881 7-902 7-923 7-943 7'963 7-983 8-002 8-020 8-0388-056 '4 2 3 5 7 8 10 12 U 15 17! 18 10 50 8-074 8-091 8-108 8-124 8-140 8-156 8-172 8-187 8-2028-217 '3 I 3 4 5 7 8 9 ii 12 3 IS 11 8. 10 23H 2457 2597 2735 2871 3005 3137 3266 3394 3520 2'0 II 22 33 45 56 67 78 89 IOO 112 123 50 10 3644 3766 3887 4005 4123 4238 4352 4465 4576 4685 2'0 IO 19 29 38 48 58 67 77 86 96 105 40 20 4793 4900 5OO6 5110 5213 534 5415 5514 5612 5709 17 8 17 25 34 42 5o 59 67 76 84 93 30 30 5805 5899 5993 6086 6177 6268 6358 6446 6534 6621 !'5 7 15 22 3 37 45 5 2 60 67 751 82 20 40 6707 6792 6876 6959 7042 7123 7204 7284 7364 7442 I'4 7 13 20 27 34 40 47 54 61 67 74 10 50 75,0 7597 7673 7749 7824 7898 7972 8045 8117 8189 I '2 6 12 18 25 3i 37 43|49 55 62 68 10 20 8260 8330 8400 8469 8538 8606 8673 8740 8807 8873 1*1 6 II 17 2 3 28 34 4045 5i 57 62 50 10 8938 9003 9067 9131 9194 9256 9319 9380 9442 9503 ro 5 10 16 21 26 3i 36 *l 46 52 57 40 20 9563 9623 9682 974i 9800 9858 9915 9973 9-0030 0086 I'O 5 IO 14 19 24 29 34 39 43 48 53 30 30 OI42 0198 0253 0308 0362 0416 0470 0523 0576 0629 "9 4 9 13 18 22 27 3i 36 40 45 49 20 40 0681 0733 0784 0836 0887 0937 0987 1037 1087 1136 8 4 S 12 17 21 25 29 33 37 42 46 10 50 1185 1233 1282 1329 1377 1424 1472 1518 1565 1611 8 4 8 12 16 2O 23 27 3i 35 39 43 9 9- 30 1657 1702 1748 1793 1838 1882 1926 1970 2014 2057 7 4 7 II 15 18 22 26 29 33 37 40 50 10 2IOI 2144 2186 2229 2271 2313 2355 2396 2437 2478 7 3 7 10 M 17 21 24 28 3i 35 38 40 20 2519 2559 2600 2640 2680 2719 *759 2798 2837 2876 6 3 6 IO 13 16 19 23 26 29 32 36 30 30 2914 2952 2991 3028 3066 3104 3MI 3178 3215 3252 6 3 6 9 12 15 18 22 25 28 3i 34 20 40 3288 3324)3361 3396 3432 3468 3503 3538 3573 3608 6 3 6 9 12 15 18 21 24 27 29 32 10 50 3643 3677 37 i i 3746 3779 3813 3847 3880 3913 3946 6 3 6 8 II 14 17 2O 23 25 28 3i 8 9- 4-0 3979 4012 4045 4077 4109 4141 4173 4205 4237 4268 '5 3 5 8 II 13 16 19 21 24 27 29 50 10 4300 4331 4362 4393 4423 4454 4484 45H 4545 4575 "5 3 5 8 10 13 15 18 21 23 25 28 40 20 4604 4634 4664 4693 4722 4751 4780 4809 4838 4866 '5 2 5 7 10 12 H 17 19 22 24 27 30 30 4895 4923 495 i 4979 5007 5035 5063 5090 51 17! 5H5 "5 2 5 7 9 12 M 16 19 21 23 26 20 40 5 f 72 5199 5226 5 2 52 5279 536 5332 5358 5384! 54io "4 2 4 7 9 II U 16 18 2O 22 24 10 50 5436 5462 5488 5513 5539 5564 5589 5614 5639! 5664 '4 2 4 6 8 IO 12 15 17 19 21 2 3 7 9- 5-0 5689 57M 5738 5763 5787 5811 5835 5859 58831 5907 "4 2 4 6 8 10 12 H 16 19 2O 22 50 10 5930 5954 5977 6001 6024 6047 6070 6093 6ll6l 6139 "4 2 4 6 8 IO II 13 15 17 19 21 40 20 6161 6184 6206 6229 6251 6273 6295 63'7 6339 6361 4 2 4 5 7 9 II 13 15 l6] l8; 2O 30 30 6382 6404; 6425 6447 6468 6489 6510 6531 6552 6573 3 2 3 5 7 9 10 12 14 16 17 19 20 40 6594 6614 6635 6655 6676 6696 6716 6736 6756 6776 '3 2 3 5 7 8 IO 12 13 15 '7 19 10 50 6796 6816 68 3 5i 6855 6874 6894 6913 6932 6952 6971 '3 2 3 5 6 8 10 II 13 15 16; 18 -6 60 9- 6990 7009 7027 7046 7065 7083 7102 7120 7139 7157 '3 2 3 4 6 7 9 IO 12 14 15 17 50 10 7175 7193 7211 7229 7247 7265 7283 7300 73I8I7335 3 2 3 4 6 7 9 10 12 14 15 '7 40 20 7353 7370 7387 7404 742i 7438 7455 7472 7489 7506 3 3 4 6 7 8 10 II 13 I4i l6 30 30 7523 7539 7556 7572 7588 7605 7621 7637 7653 7669 *3 3 4 6 7 8 10 II 13 14 16 20 40 7685 7701 7717 7732 7748 7764 7779 7795 7810 7825 '3 3 4 5 7 8 9 ii 12 14 15 10 50 7841 7856 7871 7886 7901 7916 793i 7945 7960 7975 '2 2 4 5 6 7 9 10 II 12 14 5 70 9- 7989 8004 8018 8033 8047 8061 8075 8090 8104 8118 2 2 4 5 6 7 8 9 10 12 13 50 10 8131 8i45 8i59 8i73 8187 8200 8214 8227 8241 8254 "2 2 4 5 6 7 8 9 IO 12 U 40 20 8267 8281 8294 8307 8320 8333 8346 8359 837 1 ! 8384 *2 2 3 4 5 6 8 9 10 II 12 30 30 8397 8410 8422 8435 8447 8459 8472 8484 8496! 8508 '2j 2 3 4 5 6 8 9 IO II 12 20 40 8521 8533 8545 8557 8568 8580 8592 8604 86i5J 8627 2 2 3 4 5 6 7 8 9 IO . I I 10 50 8638 8650 8661 2673 8684 8695 8706 8718 8729; 8740 "2 2 3 4 5 5 6 7 8 9 10 4 g M M M M M M M M M M S S S s s s s s s s s I s A.M. cC 10 9 8 7 6 8 4 3 2 1 1 5 10 15(20125 30 3540 45 50 55 A.M. OB EAST OF MEBIDIAN. PABTS FOB SECONDS. For A.M. Time look for the log. next greater, and for P.M. that next less ; the difference in either case gives the seconds. Also p.m. time subtracted from 12 Hours gives a.m. time. ABBEEVIATED TABLES. (VII.) LOG-COSINES. SUBTRACT PARTS FOR MINUTES. i l i t f r t f t , / ! / / i 1 ; t / i ! 10 20 30 40 50 10 20 30 40 50 1 2 3 4 5 6 1 8 9 s u K 5-0000 oooo oooo ooco oooo 1*9999 1 49999 9999 9999 9999 9998 9998 C ) O o o o 2 4-9997 9996 9996 9995 9995 9995 3 4-9994 9993 9993 9992 999i 9990 o o c ) O o i i I i 4 4-9989 9989 9988 9987 9986 9985 5 4-9983 9982^9981 998o:9979 9977 o C ) i i I i 6 4-9976 9975 9973 9972 9971 9969 7 4-9968 9966 9964 9963 9961 9959 o ( ) I I r 2 2 8 9956 Q954 995 2 9950 9948 9 H6 9944 9942 9940 9938 9936 o O ( ) 1 I 1 2 2 10 4-9934 993 1 9929 9927 9924 9922 11 4-9919 9917 9914 9912 9909 9907 o I I 2 2 2 12 4-9904 9901 9899 9896 9893 9890 13 4-9887:9884 9881 9878 9875:9872 o I 2 2 2 3 14 4-9869 9866 9863 9859 9856 9853 15 4-9849 9846 9843 9839 9836 9832 o 2 2 2 2 3 16 4-9828 9825 9821 9817 9814 9810 17 4-9806 9802 9798 9794 9790 9786 o 2 2 3 3 3 18 4-9782 9778 9774 9770 9765 9761 19 4-9757 9752 9748 9743 9739 9734 o 2 2 3 3 3 4 20 49730 9725 9721 9716 97ii 9706 21 4-9702 9697 9692 9687 9682 9677 I 2 2 3 3 4 4 22 4-9672 9667 9661 9656 9651 9646 23 4-9640 9635 9629 9624 961819613 2 2 3 3 4 4 5 24 4-9607 9602 9596 9590 95841 9579 25 4-9573 9567 956i 9555 9549 9543 2 2 3 3 A 5 5 26 4-9537 9530 9524 9512 955 27 4-9499 9492 9486 9479 9473 9466 2 2 3 4 4 5 6 28 4-9459 9453 9446 9439 9432 9425 29 4-9418 94" 9404 9397 9390 9383 2 3 3 4 5 6 6 30 4-9375 9368 936i 9353 9346 9338 31 4-933I 9323 9315 9308 9300 9292 I 2 3 4 4 5 6 7 32 4-9284 9276 9268 9260 9252 9244 33 4-9236 9228 9219 9211 9203 9194 2 2 3 4 5 6 6 7 34 4-9186 9177 9169 9160 9151 9142 35 4-9I34 9125 9116 9107 9098 9089 2 3 4 4 5 6 7 8 36 4-9080 9070 9061 9052 9042 9033 37 4-9023 9014 9004 8995 8985 8975 2 3 4 5 6 7 8 9 38 4-8965 8955 8945 8935 8925 8915 39 4-8905 8895 8884 8874 8864 8853 2 3 4 56 7 * 9 40 4-88 43 8832 8821 8Sro 8800 8789 41 778 8767:8756 8745 8733 8722 2 3 4 5 7 8 9 10 42 48711 8699 8688 8676 8665 8653 43 4-8641 8629 '86 i 8 8606 8594 8581 2 3567 8 9 IO 44 4-8569 8557 8545 8532 8520 8507 45 495 8482 8469 8457(8444 8431 2 4567 9 10 u 46 48 4-8418 4-8255 8404 8241 839i 8227 8378 8213 8365 8198 835i 8184 47 49 4-8338 4-8169 8324 8i5S 8311 8140 8297 8125 8283(8269 8111 8096 3 3 4 5 4 6 7! 8 7 9 9 10 1 1 12 12 13 50 4-8081 8066 8050 8035 8020 8004 51 4'7 ?8 9 7973 7957 794i 7926 7910 2 3 5 6 8 9 1 1 12 14 52 4-7893 7877 7861 7844 7828 7811 53 4*7795 777* 776i 7744 7727 7710 2 3 5 7 8 10 12 I 3 15 54 4-7692 7675 7657 7640 7622 7604 55 4'7 756S 7550 7531 7513 7494 2 4 5 7 9 II 13 14 16 56 4-7476 7457 7438 7419 7400 57 361 7342 7322 7302 7282 7262 2 4 5 8 10 12 14 16 18 58 4-7242 7222 7201 7181 7160 7139 59 4-7118 7097 7076 7055 7033 7012 2 4 5 8 IO 13 15 17 19 60 4-6990 6968 6946 6923 6901 6878 61 4-6856 6833 6810 6787 6763 6740 2 4 7 9 II 14 16 18 20 62 4-6716 6692 6668 6644 6620 6595 63 4'6 =570 6546 6521 6495 6470 6444 2 5 7 10 12 15 17 20 22 64 4-6418 6392 6366 6340 6313 6286 65 4-6259 6232 6205 617716149 6121 3 5 8 ii 13 1 6 19 22 24 66 4-6093 6065 6036 6007 5978 5948 67 4-59I9 5889 5859 5828 5798 5767 3 6 9 12 15 18 21 24 27 68 4-5736 5704 5673 5641 5609 5576 69 4-5543 55-c 5477 5443 5409 5375 3 7 i 3 13 16 20 23 26 30 6 5 10 15 20 25 i 2 3 4 30 i 35 40 45 50 55 i 2 3 o 70 4-5341 5323 5306 5288 5270 5253 3 7 io( 14 5235 5217 5199 5181 5163 5M5 A 7 " 14 71 4-5126 5108 5090 5071! 5052 5034 A 7 II 15 5015 4996 4977 4958 493914919 A $ II 15 72 4-4900 4880:4861 4841 4821 4801 4 8 II 16 4781 476i 4741 4721 4700 4680 >\ 3 12 16 73 4-4659 4639 4618 4597 4576 4555 4 8 12 17 4533 4512 4491 4469 4447 4425 A 1 ? 13 17 74 4 '4403 4381 4359 4337 43M 4292 4 9 13 18 4269 4246 4223 4200 4i77 4153 S < ? !4 19 75 4-4I30 4106 4083 4059 4035 40ic 5 10 14 19 3986 396i 3937 3912 3887 3862 5 I< ^ 15 20 76 77 4-352I 3811 3786 3493 3466 376o 3734 3438 34 ic 3708 > 3382 5 6 IO n I 5 20 16 22 3682 3353 3655 3629 3325 3296 3602 3267 3575(3548 3238 3208 C i i i 16 i 17 21 23 78 4-3179 3H9 3"9 3089 3058 3027 6| 12 18 24 2996 2965 2934 2902 2870 2838 c i 3 19 25 79 4-2806 2773 2740 2707 2674 \. 264C 7 13 20 26 2606 2572 2538 2503 2468 2432 7 i. * 21 28 80 4-2397 2361 2324 2288 2251 2214 7 15 22 29 2176 2138 2100 2061 2O22 1983 S i 5 23 3 1 81 4-1943 1903 1863 1822 1781 I73S c^ 16 24 33 1697 1655 1612 1568 1525 1480 5 i 7 26 35 82 4-1436 1390 1345 1299 1252 I2OC ( . 18 28 37 "57 1109 1060 ion 0961 0910 1C 20 30 39 83 4-0859 0807 755 0702 06 4 5 0594 ii 21 32 43 0539 0483 0426 0369 0311 0252 u 23 34 46 84 4-0192 0132 0070 0008 3'994f 3-988J 12 25 37 50 5-9816 975 9682 9614:9545 9475 M 27 41 55 85 3-9403 9330 9256 9181 9104 L 902(1 15 30 45 60 8946 8865 8783 8699 8613 8525 .17! 34 5 67 86 3-8436 8345 8251 8156 8059 795$ 1C 38 59 76 7857 7752 7645 7535 74 2 3 7307 22 441 66 88 87 3-7188 7066 6940 6810 6677 6535 z( S 2 78104 6397 6250 6097 5939 J5776 5605 32 63; 95 126 88 3-5428 5243^5050 4848 4637 44M 41 81 122 l62 4179 393i 3668 3388 3088 2766 57;"3i69 226 89 3-2419 2041 1627 1169 0658 0078 92183 295396 -9408 8617 7648 6398 4637 1627 250500750 1000 ABBEEVIATED TABLES TEAVERSE TABLE. (VIII) DIFF. LAT. AND DEP. FOE DIST. I'.- 1 j 2 3 4 i 5 6 7 8 9 Co. Co. DEO. D.lat Dep. D.lat Dep. D.lat Dep. D.la t Dep. D.lat Dep. D.lat Dep. D.lat Dep. D.lat E )ep. Mat Dep. D.lal Dep. DEC. OO 'OO I'OO 02 I'OO 03 roc > -05 roo 07 COO 09 '99 io 99 12 99 14 99 16 6 10 98 -17 98 19 98 -21 97 '22 97 24 97 26 9 6 28 96 29 95 '3 1 9S 33 10 20 '94 '34 93 36 93 '37 92, -39 91 4i 91 42 90 -44 89 45 88 '47 8 7 48 20 30 87 -50 86| 51 85! '53 84 '54 83 56 82 57 81 59 So 60 79 62 78 63 30 40 77 -64 75 66 74 -67 73 -68 72 69 71 71 69 72 68 73 67 74 66 75 40 50 64 77 63 78 62 79 60 -80 59 81 57 "82 56, '83 54 84 53 '85 51 -86 50 60 50 -87 48 87 47 '88 45 '89 44 90 42 -91 41 91 39 92 '37 '93 361 '93 60 70 '34 '94 33 95 31 '95 29 -96 28 96 26 -97 24 -97 22 '97 21 -98 I9i -98 70 80 17 -5 8 16 '99 14 -99 12 '99 io '99 09 I'OO 07 roo 05 1 oo 03 roo 02 I -00 80 To CONVERT DEP. INTO D. LONG. Take the lat. as Course and find D.lat. Then the dep. -f- this D.lat. = D.long. Ex.: Lat. 50 and Dep. 53'. Here Co. 50 gives D. Lat. '64 . And 53 -=- -64 = 83' D.long. and so on. j (IX.) COEE FOE SUN'S OBS. ALT. + COEE. FOE STAE'S OBS. ALT. - HEIGHT IN FEET. HEIGHT IN FEET. ALT. 5 10 15 20 25 30 35 40 45 50 55 60 ALT. 5 10 15 2 25 30 35 40 45 50 55 60 6 5 5 4 4 3 3 2 2 i i i 6 IO ii 12 ] 3 13 14 14 15 15 15 15 16 7 7 6 5 4 4 3 3 3 2 2 I i 7 9 io II 1 2 12 13 13 H '4 15 15 8 7 6 6 5 5 i 4 4 3 3 3 3 2 8 9 io IO 1 I II 12 12 ! 3 U 13 14 14 10 9 S 7 6 6 5 5 5 4 4 4 4 10 7 8 9 1 O 10 II II ii 12 12 12 13 15 io 9 9 8 8 7 7 6 6 6 5 15 6! 7 7 8 8 9 9 10 IO IO II II 20 II 10 10 9 9 8 8 7 7 7 6 6 20 5 I 5 6 7 7 8 8 9 9 9 10 10 25 12 II 10 IO 9 9 8 8 7 7 3 r 6 25 4 5 6 6 7 7 8 8 9 9 9 io 30 12 II II 10 10 9 9 8 8 8 7 7 30 4 ! 5 5 6 6 7 7 8 8 9 9 9 35 13 12 II 10 10 9 9 9 8 8 3 r 7 35 3 I 4 5 6 6 7 7 8 8 8 9 9 40 13 12 II 1 1 IO IO 9 8 8 8 7 7 40 3 ! 4 5 S 6 6 7 7 8 8 8 9 45 13 12 II II 10 10 10 9 9 9 8 8 45 3 4 5 5 6 6 7 7 7 8 8 8 50 13 12 II II IO IO IO 9 9 9 9 8 50 3 ! 4 5 5 6 6 7 7 7 8 8 8 60 13 12 12 11 IO IO IO 9 9 9 * j 8 60 3 4 4 5 5 6 6 7 7 7 8 8 70 13 *3 12 II II 10 10 9 9 9 8 8 70 2 3 4 5 5 6 6 6 7 7 8 8 80 14 13 12 II II 10 IO IO 9 9 ? J 8 80 2 3 4 4 5 5 6 6 7 7 7 8 (X.) FOE CONVEETING AEG INTO TIME, AND TIME INTO AEG. O i o o o o o / / ^ / , / / / DEC 1 2 3 4 5 6 7 8 9 DEC H. M | IT M. H. M. H. M. H . M. H. M. H. M. H. M. H. M. H. M M. S. M . S. M. S. M. S. M . s. M. S. M. S. If. S. M. S. M. S. . O O 4 8 12 O 16 2O o 24 28 o 3* o 36 10 o 40 O 44 o 48 o 5* 56 i o i 4 I I J I 12 I 16 10 20 I 2O I 24 I 28 i 3 2 I 36 I 40 i 44 I 48 I 52 I 56 20 30 2 2 4 2 8 2 12 z 16 2 2O 2 24 2 28 ! 2 32 * 36 30 40 2 40 2 44 2 48 2 ^2 2 56 3 o 3 4 3 * 5 3 i* 3 '6 40 50 3 2O 3 24 3 28 3 3 2 3 36 3 43 3 44 3 48 3 5 2 3 5 6 50 60 4 o 4 4 4 8 4 12 4 16 4 20 4 24 4 28 i 4 32 4 36 60 70 4 40 4 44 4 48 4 5* 4 56 c o 5 4 5 *" 5 12. 5 16 70 80 c 2C 5 24 5 28 5 3* 5 36 5 40 5 44 5 48 5 5 2 5 56 80 If the degrees exceed 90, find the time for those in excess and add 6 hours. Also add 1 sec. of time for each i' - Diff. Lat. and Dep. for distances 1' to 31' may be found without multiplication by Tab. HA. in the same way as in finding the Longitude Correction, by entering with Diff . Lat. atid Dep. for 1' at the side and the Dist. at the top. FOE COEEECTING THE DECLINATION AND EQN. OF TIME. (XI.) GREENWICH TIME FROM NOON. MINUTES. H.D H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 H 9 H 10 H 11 H 12 M 10 M 20 M 30 M 40 M 50 // / // / // / /'/ / // / // / // / // / // / // / // / // / // // II // II H 1 O'l 0'2 0*3 0-4 o'5 o'6 07 c-8 0-9 O'lO O'll 0'12 I i i 2 O'2 o*4 0-6 0-8 O'lO 0*12 0-14 0-16 0-18 0'20 0'22 0-24 o I I i i 3 0-3 0-6 0-9 0'12 0-15 0-18 O'2I O-24 0*27 O'3O o'33 0-36 I I 2 2 4 0'4 0-8 O'I2 0-16 0'20 0-24 % 28 0-32 0-36 0-40 0-44 0-48 I 2 3 3 5 0'5 O'lO 0-15 O'2O 0-25 0*30 0'35 0-40 o'45 o'So o'55 I'OO 2 2 3 4 6 0-6 0'12 0-18 O'24 0*30 0-36 O'42 0-48 o'54 Q 1-6 ri2 2 3 4 ! 5 7 07 0'14 0'2I 0-28 0'35 0-42 0-49 0-56 '3 io 1-17 1-24 2 3 4 5 8 0-8 O"i6 0-24 0-32 0-40 0-48 0-56 4 12 '2O i 28 r36 3 4 5 6 9 0-9 0-18 0-27 0-36 0'45 o'54 I '3 12 '21 30 i'39 1-48 3 4 6 7 10 O'lO O'2O 0-30 0*40 0*50 ro no 20 3 40 I ^0 2'O 2 3 5 7 8 11 O'll O"22 0-33 0-44 o'55 1-6 i 17 28 '39 I*5O j 2' I 2'12 2 4 6 8 9 12 0'12 0-24 0*36 0-48 ro ri2 1-24 36 48 2'0 2'12 2-24 2 4 6 8 io 13 o'i 3 O'26 0-39 0-52 i'S 18 1-31 "44 i '57 2'IO 2-23 2-36 2 4 6 8 i io 14 0-14 0-28 0-42 0-56 no -24 1-38 52 2'6 2 '20 2'34 2-48 2 5 7 9 ii 15 0-15 0-30 o'45 I'D i'i5 30 i'45 2'O 2-15 2-30 2'45 3 -0 2 5 7 10 12 16 0*16 0-32 0-48 1*4 I'2O 36 1-52 2'8 2-24 2'4O 2-56 3'12 3 5 8 1 1 13 17 0-17 o'34 0-51 r8 I-2 5 42 i '59 2-16 2'33 2-50 37 3'24 3 6 8 ii M 18 0-18 0-36 o'54 1*12 1-30 48 2-6 2-24 2-42 3*o 3-i8 3-36 3 6 9 12 15 19 0-19 0-38 0-57 1-16 1-35 '54 2*13 2-32 2-51 3-10 3-29 3^8 3 69 12 15 20 O"2O O'4O I'D I'2O I-40 2'0 2 '20 2-40 3-0 3 '20 3-40 4'0 3 j 7 io 13 16 21 O'2I 0-42 1-3 I-2 4 i*45 2-6 2-27 2-48 3 '9 3'3o 3-51 4'12 3 7 !0 H 16 22 0-22 0'44 1-6 1-28 1-50 2'12 2'34 2-56 3'i8 3-40; 4-2 4-24 4 7 ii 15 18 23 0*23 0-46 '9 1-32 1-55 2-18 2- 4 I 3 '4 3' 2 7 3*50 4-13 4'36 4 8 ii 15 19 24 0-24 0-48 '12 1-36 2'O 2'24 2-48 i 3-12 3-36 4-0 i 4-24 4-48 4 8 12 16 20 25 0-25 0'50 15 I-40 2'5 2*30 2'55 3*20 3'45 4-10 4'35 5-0 4 8 12 16 20 26 O*26 0-52 -18 I'44 2 % IO 2- 3 6 3-2 3-28 3 '54 4-20 4-46 5'12 4 9 13 17 21 27 0-27 0'54 -21 I- 4 8 2-15 2-42 3 '9 3-36 4*3 4'30 4'57 5 '24 4 9 3 18 22 28 0-28 0-56 -24 I- 5 2 2'20 2-48 3'i6 3 '44 4-12 4-40 : 5 -8 5-36 5 9 H 19 23 29 0-29 0'58 P27 I- 5 6 2'25 2'54 3-23 3'5* 4-21 4'5o 5'i9 5'48 5 io 14 19 2 4 30 0-30 i-o 1-30 2'0 2-30 3'0 3'3o 4-0 4'3o S'o 5*30 6-0 5 10 IS 20 2 5 31 0-3I 1-2 "33 2*4 2'35 3'6 3'37 4'8 4'39 5'io 5-4i 6-12 5 10 15 21 26 32 0*32 (1*4 '36 2'8 2*40 3-12 3-44 4-16 4-48 5 '20 5'52 6-24 5 II 16 21 2 7 33 0'33 i -6 -39 2'12 2'45 3'i8 3-51 J4-24 4-57 5 '30 6-3 6-36 5 II 16 22 ] 27 34 o'34 1-8 -42 2-16 2-50 3'24 3-58 432 5'6 5-40 i 6-14 6-48 6 ii 17 23 28 35 o-35 i -io | '45 2'20 2'55 3'3o 4'5 4-40 5-I5 5-501 6-25 7-0 6 12 17 23 29 36 0-36 i -12 | -48 2-24 3-0 3-36 4-12 4-48 5' 2 4 6'o 6*36 7-12 6 12 18 24 30 37 0'37 IPI 4i '5 1 2-28 3'5 3 '42 4-19 4-56 5*33 6'io 6-47 7-24 6 12 18 25 11 38 0-38 n6 i -54 2'32 3-10 3H8 4-26 5-4 5-42 6-20 6-58 7-36 6 13 19 25 32 39 0-39 ri8i -57 2-36 3'i5 3'54 4'33;5*i2 5'5i 6-30 7'9 7-48 6 13 1 9 26 32 40 0-40 I '20 2'0 2*4O 3'2o 4-0 4.40 5-20 6-0 6*40 7'2O 8-0 7 13 20 27 33 41 0-41 1*22 2'3 2'44 3-25 4-6 4'47 5-28 6-9 6-50 7-31 8-12 7 14 20 27 34 42 0*42 j i '24 I 2"6 2-48 3 '30 4-12 4'54 5-36 6-18 7-0 7-42 8-24 7 14 21 28 35 43 0'43 1-26 2-9 2'52 3'35 4-18 5'i 5'44 6-27 7'io 7'53 8*36 7 I 4 21 29 36 44 O"44 I-28 2'12 2-56 3-40 4-24 5'8 5-52 6-36 7-20 8'4 8-48 7 15 i 22 29 37 45 0-45 1 1-30 2-15 3'0 3'45 4'30 5'i5 6-0 6'45 7-30 i 8-15 9-0 7 1 5 22 3o 37 46 0-46 1 1-32 2-18 3*4 3*50 4*3 6 5-22 6-8 6'54 7-40 8-26 9-12 8 15 2 3 3i 38 47 0-47 1-34 2-21 3-8 3'55 4-42 5-29 6-16 7'3 7-5o 8-37 9-24 8 16 23 3i 39 48 0-48 1-36 2-24 3-12 4-0 4-48 5-36 6-24 7-12 8-0 8-48 9'36 8 16 ! 24 32 40 49 0-49 1-38 2-27 3'i6 4'5 4'54 5'43 6-32 7-21 8-10 8-59 9-48 8 16 24 33 4i 50 0*50 1*40 2*30 j 3 '20 4-10 5'0 5'5o 6*40 7-30 8-20 9-10 IO'O 8 17 25 33 42 51 0-51 I -42; 2-33 3-24 4-15 5'6 5'57 6-48 7'39 8-30; 9-21 IO'I2 8 17 25 34 42 52 0-52 1-44 2-36 3-28 4-20 5-12 6-4 6-56 7-48 8-40 9-32 IQ'24 9 17 26 35 43 53 0'53 1-46 2-39 3'32 4'25 5-18 6-ii 7 '4 7'57 8-50 9'43 10-36 9 18 26 35 ! 44 54 0-54 1-48 2-42 3-36 4'30 5-24 6-18 7-12 8-6 9-0 9'54 10-48 9 18 27 36 45 55 o'55 1-50 2-45 3 '40 4'35 S'30 6-25 7-20 8-15 9-10 10-5 I I'D 9 18 27 37 46 56 0*56 1-52 2-48 3*44 4-40 5-36 6-32 7-28 8-24 9-20 10-16 11-12 9 19 28 37 47 57 0-57 i'54 2-51 3-48 4'45 5-42 6'39 7-36 8-33 9 '30 IO'27 11-24 9 19 28 3 47 58 0-58 1-56 2-54 3'52 4-50 5-48 6-46 7*44 8-42 9-40 10*38 11-36 10 19 2 9 39 48 59 o'59 1-58 2-57 3-56 4-55 5'54 6-53 7'5* 8-51 9-50 10-49 11-48 10 20 29 39 49 60 o'6o 2'O 3-0 4-0 5'o 6-0 7-0 8-0 9-0 lO'O I I'D I2'O IO 20 30 40 5o 43 EXAMPLES To show the reliability of the foregoing Abbreviated Tables : vide also Notes next page. FINDING THE TIME. Ex. I. By Inman's Method. Ex. I. By,Norie, &c., modified. (Lat. and Dec. of same name.) (Lat. and Dec. of same name.) Lat. 4023'N. -1182 ... log. sec. \ Lat. 40 23' 1182... log. sec. Dec. 10 7 N. -0068... ,, ,, P.D. 79 53 0068... ,, ,, (dec.) AH 9Q 1R Diff. 30 16 JSULvt &J J.O Z.D. 60 42 Sum 149 34 Sum 90 58 4-8531 ... -log. Hav. -sum 74 47 4-4191 ... log. cosine Diff. 30 26 4-4191 ... ,, *Kemr. 44 31 4-8531... ,, Log Hav. 9-3972 Log. Hav. 9-3972 H. M. S. H. M. S. H.A. ... 3 59 47 H.A. . . 3 59 47 Ex. II. By Inman. (Lat. and Dec. of contrary names.) Ex. II. By Norie, modified. (Lat. and Dec. of contrary names.) Lat. Dec. Sum Z.D. 38 5 30'N. 15 S. 1065., 0018., .log. it sec. 43 70 45 21 Sum 114 6 4-9238... $-log. Hav. Diff. 26 36 4-3618... Lat. 38 30' P.D. 95 15 Alt. 19 39 1065.. 0018... 4-3618.. 4-9238.. log. sec. M (dec.) log. cos. log. cos. Sum 153 24 -sum 76 42 *Remr. 32 57 Log. Hav. 9-3939 H. M. S. H.A. ... 3 58 47 Log. Hav. 9-3939 H. M. S. H.A. ... 3 58 47 * In the examples on the right-hand side of the page, the remainder is found by subtracting the half sum from the Altitude increased by 90. Thus in Example I. it is subtracted from 119 18', and in II. from 109 39'. To do this we proceed in the usual way till the last figure is reached, then borrow 9 instead of 10. THE SECANT METHOD. Add together the Log. Secants of the Latitude and declination (A) And those of the half sum and remainder ... ... (B) Then 10 + Log. A. -- Log B. - Log. Hav. H.A. EXAMPLES. Ex. I. (above). 40 23' -1182. ..log. sec. 79 53 -0068... (dec.) 29 18 Ex. II. (above). 149 34 74 47 44 31 1250 (A.) 5809. ..log. sec. 1469... 7278 (B.) Log. Hav. 9-3972 (A. B.) H. M. S. H.A. ... 3 59 47 38 30' 95 15 19 39 153 24 76 42 32 57 1065 ... log. sec. 0018... ,, (dec.) 1083 (A.) 6382... log. sec. 0762 ... 7144 (B.) Log. Hav. H. M. s. H.A. ... 3 58 47 3939 (A. B.) These results are identical with those found as above, and all the Log. Secants are taken from the same page. 44 NOTES. I The H.A. found as above is P.M. time if the Sun be west of Meridian, or what it wants of 24 hours if east. A.M. time may also be taken from the right-hand side of the Table of log. haversines. The above methods are equally applicable to Stars. II. THE DEGEEE OF DEPENDENCE. The error in the Time arising from using only four places of decimals in the logs, may be easily found as follows : When taking out the H.A. from Table VI., look among the parts for seconds in the same line for the number 10, or the nearest number to 10, and take out the seconds corresponding to it. These divided by 10 will give the error due to an error of 1 in the fourth figure of the log. vers. H.A. Thus for 2h. 20m. 10 gives 10 sec., .'.the error is 1 sec. ,, ,, 3h. 20m. 10 gives 15 sec., /.the error is 1'5 sec., &c. As the final figure in each log. is between and '5 either in excess or defect of what it ought to be, the errors will generally be found to cancel each other, and as a matter of fact will, in the aggregate, seldom exceed unity ; and, as will appear from above, the resulting error in the H.A. will therefore rarely exceed 2 seconds of time or half-a-minute of longitude. III. THE ALTITUDE-AZIMUTH. (By THE SECANT METHOD.) Lat. 40 23' "1182 ... sec. Alt. 29 18 -0595 ... P.D. 79 53 149 34 1777 (A.) Sum 74 47 "5809 Diff. 5 6 -0018 (B.) A B -log. Hav. 4-7975 Az. : S. 77 42' E. The above short method will be useful when the Azimuth is required to a greater degree of exactness than can be found by the ordinary tables, and is independent of the time. * To find the diff. subtract the half sum from the Polar distance or the Polar distance from the half sum if the latter is the greater. 45 EXPLANATION OF TABLES IV. TO XI. TABLE IV. EEQUIRED THE LOG. SEC. of 30 25' . Log. sec. 30 20' ='0639, and the parts for 5' are 4, adding which we have '0643 for the required log. sec. Conversely : To find the degrees corresponding to log. sec. '0643. The log. sec. next less is '0639, which gives 30 20', and the diff., 4, gives 5'. .-. The arc required is 30 25'. ' When the parts for 1' exceed 1 those for ' , ' may be found by dividing the parts for 1' by 2 or 4 as the case may be. When less than 1 they may be disregarded. TABLES V. and VII. are used in the same way, except, that as the cosines decrease the parts for minutes must be subtracted instead of added. TABLE VI. To TAKE OUT THE TIME FOR LOG. HAV. 9'1352. The log next less is 9'1329, which gives 2 h. 53 m., and the remainder is 23, which looked for in the same line gives 30 sec. at the top. /. The time is 2 h. 53 m. 30 sec. CONVERSELY. To find the log. hav. of 2 h. 53 m. 30 sec., 2 h. 53 m. gives 9'1329, and the parts for 30 sees, are 23, adding which we have 9'1352, and so on. When the remainder is not found exactly among the parts take the seconds corresponding to the nearest or the mean as the case may be. Thus, if, in the preceding case, the remainder had been 25, the mean of the seconds would be 32'5, or 32 to the nearest second. If it had been 24 we might have taken 30 sec., or, if 26, 35 sec., and so on. Again, suppose in the above case the remainder to be 38, we see that the next less is 35, which gives 45 sec., and remainder 3. Now in the same line 31 gives 40. .'. 3'1 gives 4 sec.; adding this, the seconds for remainder 38 will be 49, and so on. TABLE VII. It will be seen that in the last two or three lines the log. cosines decrease very rapidly, so that it will be better to find the parts by taking the diff. of the two log. cosines and dividing by 5. Example : Find log. cos. 88 8' : Here 88 5' = 3'5243 3'5243 And 88 10' = 3'5050 38'6 x 3/ 115 5 ) 193 Required log. = 3*5128 Pts.forl'= 38-6 This is seldom required in actual practice. 46 TABLE VIII. TEAVEBSE TABLE. This Table is intended to supply the place of the Traverse Table as far as it is required in the methods contained in this book. Ex. : Eequired the diff. lat. and dep. made by a ship sailing S. 35, W. 20'. Here co. 35 and dist. 1' give '82 d. lat. and '57 dep. .*. Diff. lat. = 20 x -82 = 16'4, and dep. = 20 x '57 == 11-4. Diff. of longitude corresponding to dep. may be found by dividing the Dep. by the D. Lat. which corresponds to the degree of latitude taken out as a course. Ex. : Given Lat. 42 and Dep. 12' , find D. Long. Here Lat. 42 as course gives D. Lat. '74. And 12 + -74 = 1200 -h 74 = 16'- D. Long. Conversely, D. Long, may be converted into Dep. by multi- plying by the D. Lat. taken out as above."'" Table IX. requires no explanation. Table X. When entering this table with degrees the corres- ponding time will be hours and minutes, and when with minutes it will be minutes and seconds of time. Ex. I: Convert 38 25' into time. H. M. Here 38 = 2 32 And 25' = 1 40 Ex. II : Convert 53 32' 45" into time. H. M. Here 53 = 3 32 32' = 28 45"= 3 38 25' = 2 33 40 53 32' 45"--- 3 34 11 The above is easily done by inspection. Conversely, Convert 2h. 33m. 40s. into arc, and convert 3h. 34m. 11s. into arc. H. M. Here 3 32 - 53 25' 28= 32/ H. M. Here 2 32 38 C 1 40 = 2 33 40 38 25 8 = 3 = 3 34 11 = 45' 53 32 45 TABLE XL FOE COEEECTING THE DECLINATION AND EQN. OF TIME. Ex. I. : G.Time 7h. 30m., after Ex. II. : Gr. Time 4h. 45m. Noon, H.D. 37/ + before Noon, H.D. 4S// + H.D. 37" and 7h. = 4' 19" Parts for 30m. = 18" Correction 4 37 + H.D. 43 and Parts for 45m. .'. Correction 4h. = 2 52' 32 = 3 24 - N.B. If the declination is required in an observation taken after noon it is corrected on, and if before noon, back, which accounts for the sign being reversed in Ex. II. TO COEEECT THE EQUATION OF TIME. Ex. I. : G.Time, after Noon, Ex. II. : G.Time, 3h. 40m., befpre 7h. 50m. H.D. -25s. + Noon, H.D. -88s. - H.D. and 44 and 3 40 = 2'41"=161" .-. -44 = l'61s. (Dividing by 100) And -88 = 3-22s. + H.D. 25" and 7-50 = 3' 15" - 195" .-. Dividing each by 100 We have '25s. = l'95s. + The sign of the A.M. correction is changed, as in the case of the Declination. * This multiplication and division may be performed by Table HA. 47 MULTIPLICATION AND DIVISION BY TABLES I. & II. Look for the multiplicand in the first column and the multiplier in the second top line, and note the degrees of bearing arid latitude adjacent to each. This latitude and bearing will then give the product to two places of decimals. Example I. Multiply '29 by 1'56. 29 -bearing 74, and 1-56 = latitude 50, .-. latitude 50 and bearing 74 ='44. If the number had been 29, the result would have been 44 ; or if 2'9, 4'4, and so on. Example II. Multiply '31 by 1-56. 31 = bearing 73, and 1-56 = latitude 50. Latitude 50 and bearing 73 = '47. When the numbers are not found exactly, take the mean. Example III. Multiply '38 by 1-59. Here -38 = bearing 69, and 1-59 = latitude 51. Latitude 51 and bearing 69= '61, &c. DIVISION. Is performed in exactly the reverse way. Example I. Divide '44 by 1-56. As before, 1-56 = latitude 50. Latitude 50 and *44, same column, bearing 74 = '29. If the number had been 44, the quotient would be 29, by shifting the points as in multiplication. Example II. Divide -58 by 1-59.* Here 1-59 = latitude 51, And latitude 51 and '58, same column, = bearing 70 = '36. If the exact numbers are not found, take the mean. Table I. is used in the same way, except that we use time instead of bearing ; and we use this Table in preference to Table II. when both numbers consist of two figures only. * If the divisor is too large for the scope of the Table, divide both it and the dividend by 2 or 3. Thus if we have to divide ! L6 by 3'18, we have '58-=-l'59 or '36 to the nearest whole number, &c, ALTITUDE-AZIMUTH TABLE,- To be used in combination with the Traverse Table. LATITUDE. ALTITUDE. LAT. A B LAT. A B ALT. C ALT. C ALT. C o 1 100 2 33 119 65 1 100 34 83 66 41 2 100 3 34 121 67 2 100 35 82 67 39 3 100 5 35 122 70 3 100 36 81 68 37 4 100 7 36 124 73 4 100 37 80 69 36 5 100 9 37 125 75 5 100 38 79 70 ' 34 6 101 11 38 127 78 6 99 39 78 71 j 33 7 101 12 39 129 81 7 99 40 77 72 31 8 101 14 40 131 84 8 99 41 75 73 29 9 101 16 41 133 87 9 99 42 74 74 i 28 10 101 18 42 135 90 10 98 43 73 75 26 11 101 19 43 137 93 11 98 44 72 76 | 24 12 102 21 44 139 97 12 98 45 71 77 22 13 103 23 45 141 100 13 97 46 69 78 21 14 103 25 46 144 104 14 97 47 68 79 19 15 104 ^7 47 147 107 15 97 48 67 80 17 16 104 29 48 149 111 16 96 49 66 81 16 17 105 31 49 152 115 17 96 50 64 82 14 18 105 32 50 156 119 18 95 51 63 83 12 19 106 34 51 158 123 19 95 52 62 84 10 20 106 36 52 162 128 20 I 94 53 60 85 9 21 107 38 53 166 133 21 1 93 54 59 86 7 22 108 40 54 170 138 22 93 55 57 87 5 23 109 42 55 174 143 23 92 56 5G 89 3 24 109 45 56 179 148 24 91 57 54 89 2 25 110 47 57 184 154 25 91 58 53 90 26 111 49 58 189 160 26 90 59 51 27 112 51 59 194 166 27 89 60 1 50 28 113 53 60 200 173 28 88 61 48 29 114 55 61 206 180 29 87 62 47 30 115 58 62 213 188 30 87 63 45 31 117 60 63 220 196 31 86 64 44 32 118 62 64 228 205 32 85 65 42 33 119 65 65 237 215 33 84 66 41 34 121 67 66 246 225 34 33 67 39 LAT. A B LAT. A B ALT. C ALT. C ALT. C LATITUDE. ALTITUDE. To FIND THE AZIMUTH. Take out A and B for lat. and C for alt., and with A and B as dist. and dec. and alt. as course find dep. (Trav. Tab.). Take the diff . or sum of these deps. according as lat. and dec. are of the same or contrary names ; the course corresponding to C as dist., and this sum or diff. will be the Azimuth or Bearing from South in North lat. and vice versa. EXCEPTION : If the first dep. is greater than the second, when lat. and dec. are of the same name, reckon the Bearing from North in North lat I and from South in South lat. * Reprinted from earlier editions of this work, and here inserted for the convenience of those accustomed to this method of obtaining the Azimuth. 49 EXAMPLES. 1. Lat. 50 N., Alt. 30 W'ly, Dec. 20 N. We have A. 156, B. 119, C. 87. Dist. Co. Dep. 156 and 20 -53 -4 119 30 = 59-5 CO. 87 and d. lat. 6'1 = 86 Ans. Azimuth S. 86 W. 3. Lat. 52 N., Alt. 7 E'ly, Dec. 15 N. A. 162, B. 128, C. 99. Dist. Co. Dep. 162 and 15 =-41-9 (Gr.) 128 7 = 15'6 co. 2. Lat. 40 N., Alt. 12 E'ly Dec. 8 S. By Tab. A. 131, B. 84, C. 98. Dist. Co. Dep. 131 and 8=a8'2 84 12= 17-5 co. 98 and d. lat. 35-7 = 68| Ans. Azimuth S. 68 E. 4. Lat. 45 N., Alt. 31 W'ly Dec. 15 N. A. 141, B. 100, C. 86. Dist. Co. Dep. 141 and 15 = 36-5 100 31 = 51-5 86 and d. lat. 15-0 = 80 Ans. Azimuth S. 80 W. 99 and d. lat. Ans. Azimuth N 74 E. Example 3 shows the exceptional case, the first dep. being greater than the second. The above results are the same as by actual calculation, within a quarter of a degree or so. TO FIND THE APPROXIMATE SHIP TlME. Transpose altitude and declination, and proceed as in finding the Azimuth, observing to take out C for the declination instead of altitude. Example 1 (above}. A. 156, B. 119, C. 94. Dist. Co. Dep. 156 and 30 -78-0 119 , 20 = 40-7* 94 and d. lat. 37'3=( Ans. H.A. 4h. 26m. Example 2 (above) A. 131, B. 84, C. 99. Dist. Co. Dep. 131 and 12 = 27'2 84 8-ll'7 99 and d. lat. 38'9 = Ans. H.A. 4h. 28m. Example 1 gives the H.A. within 1m. 11s., and Example 2 within 19 sec., thus affording a ready way of correcting the ship's clock when rapidly changing the longitude. The observations should not be taken when the sun is within two or three points of the Meridian. * If the second dep. is greater than the first, when lat. and dec. are of the same name, subtract the H.A, from 12 hours. 50 TO IDENTIFY AN UNKNOWN BEIGHT STAE. If, on a cloudy night, a bright star appeared for a short time through an opening in the clouds, and we wished to ascertain its name, we could do -so as follows : (1) Observe the star's altitude, and bearing by azimuth compass, to which apply the usual corrections. (2) Convert the bearing into time, and consider it as an H.A. ; also consider the altitude as declination of the same name as the latitude. (3) With the latitude, this hour-angle, and declination, find the bearing by Tables I. and II. This will be the star's hour-angle if this bearing is of the same name as the latitude, or what it wants of 12 hours if of con- trary name. Then the Meridian E. A. -(-star's H.A., according as the star is east or west of Meridian, will be the star's E.A., and the star whose E.A. agrees with this will be the body observed. Example I. June 2nd, at 8 p.m., in lat. 32 N., a star whose altitude w r as 21 bore N. 56 E. (true). Esquired its name. h. m. h. m. Bearing 56 = 3 44 ... Tab. X. S.M.T. ... 8 Lat. 32 and 3 44 = '42 S. Tab. I. Sid. Time 4 40 Dec. 21 3 44=-47N. Mer. E.A. 12 40 By Tab. II. Lat. 32 and -05 N. = N. 88 E. = * H.A. 5 52 E. Star's R.A. 18 32* Now as the E.A. of Vega is 18h. 34m., it shows that this must have been the star observed. Example II. August 22nd, at 7.13 p.m., in lat. 30 N., a star whose altitude was 20 bore S. 63| W. (true). Find its name. h. m. Bearing 63^ = 4- 14... Tab. X. S.M.T. 713 30 f <..,// -29N.f Sid. Time 10 2 20 / ' \ -40 N. 17 15 30 and -69N.=N. 59 W. = *H.A. ... 3 56 W. Star's E.A. 13 19 As Spica has the same E.A., nearly, it was the star observed. 5 1 Ursae Majoris having the same E.A., any uncertainty may be removed by finding the true bearing, using lat. 30, H.A. 3 '56 and the declination of either star, then, if this bearing agrees with the observed bearing, it shows that star to be the right one, but if not, it must be the other. The same remark applies to Capella and Eigel, the E.A. of which is 5*10. * If the sum of the Mer. E.A. and Star's H.A. exceed 24 hrs., reject 24 hrs. ; and if the H.A. (West) exceed the Mer. E.A. increase the latter by 24 hrs. fAs the angle between N. and S. 63 W., used as an H.A., exceeds 90, or 6 hrs., we mark both numbers with the same name as the latitude; vide exceptional case, p. 17. 51 TABLES FOR FINDING THE STARS. SIDEREAL TIME. FOUR MINUTES ARE TO BE ADDED FOR EACH INTERMEDIATE DAY. DAY. JAN. H. M. 1839 18 59 1918 19 38 1958 2018 FEB. H. M. 2041 21 1 21 21 21 40 22 22 20 MAR. APL. MAY. JUNE JULY H. M. 2236 22 55 23 15 23 35 23 54 14 H. M. 038 57 1 17 137 1 57 2 16 H. M. 236 2 56 3 15 3 35 355 4 15 H. M. 438 4 58 5 18 5 37 557 6 17 H. M. 637 656 7 16 736 7 55 815 AUG. H. M. 839 859 918 938 958 1017 SEP. H. M. 1039 1058 11 18 11 38 11 58 12 17 OCT. H. M. 1239 1259 13 19 1338 1358 14 18 NOV. DEC. H. M. H. M 14 41 15 1 15 21 1541 16 1620 1640 1659 17 19 1739 1759 18 18 LIST OF PRINCIPAL BRIGHT STARS. NAME. aAndromedse ...... aCassiopeise ...... aUrsse Min. (Polaris ctEridaiii ...... aArietis ......... aPersei ......... Aldebaran ...... Capella ......... Rigel ......... ttColumbse ...... aOrionis ......... /3Aurigse ...... Canopus ......... Sirius ......... eCanis Majoris SCanis Minoris Castor ......... Procyon ......... Pollux ......... eArgus aHydrse Begulus aUrsse Majoris o^Oucis R.A. H. M. 4 35 3 18 6 22 6 41 6 55 7 5 7 29 7 34 7 40 8 7 8 21 9 23 10 3- 10 58 12 21 DEC. 28-35N 56- 2N 88-49N 57-42S 23- 2N 49-32N 16-19N 45-54N 8-18S 34- 7S 7-23N 44-56N 52-39S 16-35S 28-513 26-15S 32- 5N 5-28N 28-15N 47- 4S 59-13S 8-16S 12-25N 62-15N 62-35S NAME. 7Crucis /3Crucis aUrsse Majoris 5 1 Ursse Majoris Spica AUrsse Majoris yJGentauri Arcturus a 2 Centauri aCoronse Antares aTriangulus Aust. ctScorpii .., ... aOphiuchi eSagittarii Vega SSagittarii aAquilse aPavonis aCygni aGruis /SGruis aPiscis Australis (or Fomalhaut) Markab R.A. H. M. 12 26 12 42 12 50 13 20 13 20 13 44 13 57 14 11 14 33 15 31 16 24 16 39 17 27 17 31 18 18 18 34 18 50 19 46 20 18 20 38 22 2 22 37 22 53 23 DEC. 56-368 59-118 56-28N 55-24N 10-418 49-46N 59-568 19-40N 60-278 27- IN 26-148 68-518 37' 2S 12-38N 34-26S 38-42N 26-25S 8-37N 57- 2S 44-57N 49-24N 47-228 30- 78 14-43N NOTE : Meridian Eight- Ascension = Ship Mean Time + Sidereal Time. TO FIND THE PRINCIPAL STARS ABOVE THE HORIZON AT ANY GIVEN TIME. Example. What stars are above the horizon at a place on the Equator at 8 p.m. on July 23rd ? h. m. S.M.T. July 23 8 Sid. Time Mer. R.A. Star's R.A. 8 3 16 3 6 10 3 W 16 3 6 22 3 E Then all stars whose R.A.'s lie between 10'3 and 22'3 will be above the horizon at 8 p.m. Those whose R.A. is between 1 and 16-3 will be West of Meridian, and those between 16'3 and 22-3 East. Further North more stars of North Decln. will be. visible and fewer of South Decln., and vice versd. 52 ON FINDING THE STABS. BY TABLES, PAGE 51. 1. To find what Bright Stars will pass the Meridian of a place on May llth, between 8.0 and 12.0 p.m. h. m. h. m. S.M.T. 8 and 12 Sid. Time, May 11 3 15 3 15 Meridian R.A. 11 15 15 15 Then the stars whose R.A.'s lie between 11-15 and 15*15 will pass the Meridian between 8.0 and 12.0 p.m., viz. : all stars between a 1 Crucis and a 2 Centauri. 2. To find at what time Arcturus will pass the Meridian on the same date, viz. : May llth. h. m. R.A. of Arcturus 14 11 Sid. Time, May 11 3 15 Time of Mer. Pass. 10 56 If the latitude is 50'0'N. find the Mer.-Alt. of Arcturus. Lat. 50-0' N. *Dec. 19 -40 N. M.Z.D. 30 -20 Mer. Alt. 59 "40 So that if we set the index of our sextant at 59'40 / and look towards the South point of the horizon at 10'56 we shall have no difficulty in finding the star, and can then screw in the telescope and complete the observation. If a star's declination is greater than the latitude and of the same name it will pass the Meridian between the Zenith and the Elevated Pole. If therefore both lat. and dec. are North we should have to look towards the North point of the horizon. If a star's Polar Dist. is less than the latitude it is a circum- polar star, and if its declin. is greater than the co-lat, it will not rise when they are of contrary names. 3. To find what stars are above the horizon and within 4 hours of the Meridian, East and West, at 8.0 p.m. July 23rd, and therefore suitable for time observations. h. m. Ship M.T. 8 Sid. Time, July 23 8 3 Mer. R.A. 16 3 Subtracting and adding 4 hours we have 12'3 and 20'3. /. Stars whose E.A.'s lie between these times are within 4 hours of the Meridian at 8.0 p.m. Those furthest from the Meridian are most suitable for time provided that their declination does not exceed 30 or so. Thus Spica is the only suitable star West of Meridian, and SSagittarii and aAquilae East of Meridian. NOTE. Stars whose R.A. is less than the Mer. R.A. are West of Mer. and those greater East of Mer. 53 WHERE TO LOOK FOR A GIVEN STAR AT ANY TIME. Example. Being in lat. 40 N., in what part of the heavens shall I look for Capella on May 1st, at 8.0 p.m., its R.A. and Dec. being 5h. 10m and 46 N.? (1) Find its Bearing by Az. Rule. S.M. Time, May 1st 8 '0 Sid. Time 2-36 Lat. 40 | , . ( -13 S. Dec. 46 D5IB 1-05 N. Her. R.A 10-36 Star's R.A. ... 5'10 Lat. 40 and -92 N. = N.55 W. /. It bears N. 55 W. Star's H.A. ... 5'26 (2) Find its Altitude. To do this interchange Az. and H.A. and proceed as follows : Az. 55 = 3h. 40m., H.A. 5*26 = Az. 8H. Lat. 40 and 3'40 - -59 Tab. I. Lat. 40 and 81 = -19 II. (H.A. for Dec.) 3'40 and '78 = 32 Alt. The numbers are to be added unless the H.A. is greater than 6 hours or the Bearing is of a contrary name to the latitude, in which case we take the difference. To find the altitude look for the H.A. in the right-hand or Declination H.A. column, and for '78 in the same line, the altitude will be the degrees at the top of the column in which '78 is found. If then we put alt. 32 on the sextant and look towards that point of the horizon which bears N. 55 W., we shall have no difficulty in identifying the star. TO IDENTIFY A STAR BY ITS MER. ALTITUDE. Find the star's Z.D. when on the Meridian ; then if this is less than the latitude the star and the observer are on the same side of the Equator, but if greater they are on opposite sides. Example I. In lat. 49-10 N. a star's mer. alt. bearing south was 46'20 ; find its name. Obs. Alt. 46-20 S. Z.D 43-40 S. Lat 49-10 N. Dec 5-30 N.* As the dec. of Procyon was 5 '28 N. it was the star observed. Example II. In the same latitude the mer. alt. of a star, bearing south, was 35-24' ; required, its name. Obs. Alt. 35-24 S. Z.D 54-36 S. Lat. 46-20 N. Dec. ... 8-16 S. As this agrees with the dec. of aHydrae, it must have been the star required. * When lat. and zen. dist. have the same name, their sum, if less than 90, will be the dec. ; but if greater than 90, the sum subtracted from 180. 54 TO FIND A STAB'S H.A. AT RISING AND SETTING. Convert the degrees in the polar dist. into time and with this time and the latitude take out the Nr. from Tab. I. Look for this Nr. in the last column of Tab. II. and take out the Bearing. This converted into time will be the star's H.A. at rising and setting if latitude and declination are of contrary names, or what it wants of 12 hours if of the same name. Example I. Given : Lat. 35N. Dec. 12S. : to find the H.A. at rising or setting. Here P.D. 78 = 5h. 12m. By Tab. I. Lat. 35 and 5'12 = -15. By Tab. II. -15 = Bearing 82 -5 -28 /.H.A. at rising or setting = 5h.28m. Example II. Given : Lat. 45N. Dec. 15N. : to find the rising and setting H.A. Here 75 = 5h. Om. Lat. 45 and 5'0 - -27. Tab. I. 27 = 74 = 4h. 56m. II. /.H.A. = 12h. 4-56 = 7h. 4m. In Example II. the H.A. is subtracted from 12 hrs. as Lat. and Dec. are of the same name. TO FIND AT WHAT TIMES A STAR WILL RISE, CULMINATE AND SET. Find its H.A. at rising or setting as above, and the time it passes the Meridian, to which apply the H.A. at rising, etc. Subtracting for the time of rising and adding for the time of setting. Example I. At what times will Arcturus rise, culminate and set on Oct. 3rd in lat 30 N. ? (*B.A. 14h. llm. Dec. 20 N.) Here P.D. 70 = 4h. 40m. Lat. 30 and 4-40 ='21 Tab. I. And -21 - Bearing 78 = 5- 12 II. /, H.A. = 12h. -5-12 = 6-48 "R.A. 14-11 Sid. Time 12-47 Mer. Pass H.A. 1.24 p.m. ... 1.24p.m. 6.48 .. 6.48. Setting 8.12p.m.Rising6.36a.m. Answer: Arcturus rises at 6.36 a.m., culminates at 1.24 p.m., sets at 8.12 p.m. Example II. At what times will Jupiter rise, culminate and set on Feb. 16th in lat. 40 N. ? ( :;: R.A. 23h. 50m., Dec. 15 S.) Here P.D. 75 = 5h. Om. [ *R.A. 23-50 Sid. Tim 21-40 Lat. 40 and 5-0= -22 22 = 78 = 5-12 Mer. Pass 2.10 p.m. ... H.A. 5.12 2.10 p.m. 5.12 Sets 7.22 p.m. Rises 8.58 a.m. .-.Jupiter rises at 8.58 a.m., culminates at 2.10 p.m., sets at 7.22 p.m. 55 APPLICATION OP TABLES I. AND II. TO GKEAT CIKCLE SAILING. In the Time Azimuth two sides of a spherical triangle and the included angle are given to find the azimuth, and in great circle sailing the same parts are given to find the course. It is evident therefore that the operation is precisely the same. Thus, if in Ex. I., p. 17, we take lat. 40 N. as the latitude from, and dec. 20 N. as lat. 20 N., the latitude to, and the H.A. 3h. 48m. as din 7 , long., and work it out in the same way, we shall have as course S. 85J E. If the latitude of either place is greater than 60, we must proceed as in Ex. I., p. 22 ; which example might be expressed as follows : Given: Lat. A. 76 N., Lat. B. 12 N., Diff. long. 51 E. : To find the Course from A. to B. : By Tab. (i.), p. 35, Lat. 76 = Lat. 53 and Divisor 3. By Tab. I., Lat. 53 N. and H.A. 3h. 24m. ( = 51) = 1'-07 S. Also by same table Dec. 12 N. and H.A. 3h. 24m. = -27 N. Which latter divided by 3 = '09' N. Hence we have ... 1''07 S. And -09N. . ' /. Thediff = -98 S. Again by Tab. (ii.), p. 35, Lat. 76 = Lat. 44. /.By Tab. II., Lat. 44 and -98' S. = S. 55 E., the Course required. To find the approximate distance. Take from the column (a') the numbers for the co-lat. B, and din , long, and multiply them together by the Table. Look for the result in the column at the top of which is the degree denoting the course ; the corresponding bearing will be the distance required, in degrees. Thus in the above example : Co-lat. B or 78 = T-02 1 , ,. Diff. long. 51 = l'-29 } (a } C lumn ' And l'-02 x l'-29 = l'-31 by Tab. II. Lastly, Course 55, and l'-31 = Bearing 53. ,. /. The Dist. = 53 x 60, or 3,180 miles. 56 THE ALTITUDE-AZIMUTH BY TABLES I. AND II. N.B. The degrees at the top of the Tables serve both for latitude and altitude, and those in the Bearing Column, Tab. II., # #> both for zenith dist. and polar dist. KULE. From the last column of Tab. II., p. 29, take out the numbers corresponding to the zenith dist. and polar dist., which denote by A. and B. With the lat. and A. take out the Nr. from Tab. I., and with the lat. and B. take out the Nr. from Tab. II., marking the first Nr. with the opposite name to the latitude, and the second with the same name as the declination. When both names are alike, take their sum with the common name ; and when different, their difference, \vith the name of the greater. This shows the point from which to reckon the azimuth. To find the Azimuth. With the altitude as latitude and this sum or difference, take out the Nr. from Tab. II. and look for it in the last column of Tab. II., p. 29, when in the same line will be found the bearing, or azimuth which mark as above directed. Example I. Lat. 50 N., Z.D. 60, West of Her., P.D. 70 (N). Here A = -50, B = -34 Lat. 50 and "50 = '59 8. Tab. I. Lat. 50 and "34 = -52 N. Tab. II. Alt. 30 and -07 S = '08 S. Tab. II. .-. Azimuth = S. 86 W. By calculation this would be 85 47' W. Example II. Lat. 52 N., Z.D. 83, E. of Her., P.D. 75 (N). Here A = -12, B = -26 Lat. 52 and '12 = -16 S. Tab. I. Lat. 52 and -26 = -43 N. Tab. II. Alt. 7 and -27 N. = -27 N. Tab. II. .-. Azimuth = N. 74 E. By calculation this would be 74 33'. 57 Example III. Lat. 40 N., Z.D. 78, E. of Mer., P.D. 82 (S). A -21, B -14 Lat. 40 x -21 = -17 S. Tab. I. Lat. 40 x -14 = -18 S. Tab. II. Alt. 12 and '35 S. = '36. Tab. II. .-. Azimuth = S. 68 E. By calculation this would be 68 39'. After a little practice this may be abbreviated as follows : 17 S. 18 S. Alt. 12 & -35 S. = -36 = S. 68 E. THE CONSTRUCTION OF TABLE II.* The first column (for lat. 0) consists of natural co-tangents ; the line of figures at the top TOO, &c., to 2'00 are natural secants of the degrees above them, and the single line at the bottom natural tangents of the degrees at the top of the column. The other columns are the products of the natural co-tangents at the side of the Table and the natural secants at the top, and as both these are natural numbers the Table may be said to give the result of the multiplication of any two natural numbers corresponding to those given in the first column and at the top to two places of decimals. Thus to multiply 1-19 by 1/56 we have by the Table 1/85, which, to two places of decimals, is the same as by actual multiplication. Conversely 1/85 * 1*56 = 1*19, which also is the same result as that obtained by actual division. The numbers at the bottom of the Table are the natural tangents of 0-60. Those in the Dep. or d column are the natural co-secants of 10 -90 ; and in the last column on the right, the natural cosines of the same. If the natural tangent of an angle greater than 60 be required, take the nat. co-tan, of the complement ; and if a natural secant, take the natural co-secant of the complement. * This Table was first published in the 4th Edition of this little book in 1874. It was subsequently, by the Author's permission, inserted in Lecky's "Wrinkles," Inman's Nautical Tables, &c. In the former, and in The General Utility Tables, by the same Author, it now appears in an expanded form, as Table (C). 58 EXPLANATION OF THE DOUBLE CHRONOMETER RULE. C FIG (/) A FIG (2) Let C be the true zenith of the observer, CA., CB small portions of circles of equal altitude, BD a small portion of a parallel of latitude by D.R., CD a perpendicular from C on BA, or BA produced. (Fig. (1) is for observations taken on the same side of the meridian, fig. (2) for those taken on opposite sides).* Then the first observation worked with the D.R. lat. will place the ship at A, the second will place her at B. Therefore AB is the discrepancy (in dep.) between the two positions. Also CAD is equal to the azimuth at the first observation, and CBD is equal to that at the second. fig. (1) AB=BD-AD = G D Cot. B - C D Cot. A = C D (Cot. B - Cot. A) A B CD = Cot. B - Cot. A A B Sec. I Cot. B Sec. I - Cot. A Seo, 7 Diff. long. Similarly in fig. (2) CD Cot. B Sec. I - Cot. A Sec. I Diff. long. Cot. B Sec. I + Cot. A Sec. I The values of cot. B, sec. I and cot. A sec. I aretakenfromTab.il., and designated as (a) and (6). .-. Corr. for lat. = Diff-Jong. ^ taking the upper or lower sign according as the * ' + \ a ' bearings are in the same, or adjacent quadrants. * These triangles, being supposed to be very small, may be treated as plane triangles, right-angled at D. 59 The corrections for the two longitudes will be AD and BD expressed in diff. long. or, AD sec. /, and BD sec. /. But AD = CD cot. A, and BD = CD cot. B. .'. The corrections are CD cot. A sec. I, and CD cot. B sec. I \ or, corr. for lat. x (a), and corr. for lat. x (b), and it is evident by fig. (1) that when the observations are in the same quadrant, both corrections must be allowed in the same direction ; but when in adjacent quadrants, they must be allowed in opposite directions, as in fig. (2), to make the two longitudes agree. It will also be seen, by fig. (1), that if the sun bore S.E 1 * and the correction for lat. were North, that for long, would be East ; and by fig. (2), if the sun bore S.W ly - and the corr. for lat. were North, that for long, would be West. Hence the Rule on pp. 7 and 8 : S. E. S. W. N. W. N X E. THE TIME AZIMUTH. In a spherical triangle ZPS, where PZ = 90 I, PS = 90 + d, ZPS = /i, and PZb = A, the azimuth, it may be shown that Cot. PS sin. PZ = cot. A. sin. ZPS -f cos. PZ cos. ZPS. Or, Cot. A. sin. ZPS. = cot. PS. sin. PZ -cos. PZ cos. ZPS. Cot. A. sin. h = tan. d cos. I - sin. I cos. h Whence, Cot. A. sec. I = tan. d cosec. h tan. I cot. h Or, Cot. A. sec. I = tan. d cosec. h -tan. I cot. h (0) when latitude and declination are of opposite names. To adapt this to Table I., which contains the products of natural tangents and natural co-tangents, we assume that cosec. A = cot. h' ; then tan. d cosec. h = tan. d cot. h' which is a similar expression to the second term of the right-hand side of the equation (0), and shows that both the numbers for the latitude and declination may be taken from the same table, and without any sacrifice of accuracy, while at the same time the table is applicable to all declinations from to 58, and may therefore be used, not only in finding the bearing of tne sun, Out, aho of all stars within the above limits. 60 Explanation of the Ex- Meridian. In the spherical triangle ZPS, where ZP represents the co-latitude, PS the polar-distance, ZS the zenith-distance, and ZPS the hour-angle, (h), supposed to be very small, it may be shown by spherical trigonometry, that Vers. h cos. I cos. d. But (Z~"d) = the mer. zen. dist. = s', suppose, cos. z' - cos. z ' Vers.fc- cos. I cos. d Or cos. I cos. d Vers. fo = cos. z' cos. a = 2sin.*J_l' sin/ _- i '...(A) But, since the sun is supposed to be near the meridian, z4-z' J5~ = z y nearly ; and 2-3' = the reduction, = C, suppose, /^ /. From (A), 2. cos. d cos. / hav. h = 2 sin. 2, sin. . . . (B). But since _ is very small, we have, by using the circular measure, Sin. = "2 2r From (B), cos. d cos. I hav. h sin. z. ~ /. C = 2 r cos. d cos. I cosec. z hav. A = 2 r hav. /i cos. d cos. I sec. alt. Where r = 57 28', or 3438'. The upper part of the Table gives the values of cos. I sec. alt., or N. ; and the lower part those of 2r. hav. h x N. A further correction (for the declination) may be applied when both the reduction and the declination are considerable. NOTE. The Table exhibits at a glance the values of the reduction corresponding to any given value of N, and therefore shows the error that would be produced by an error of 1 minute (or any other portion) of time, in the hour-angle, which is important as showing the degree of ^dependence that may be placed in a given observation. 61 To find by Table II. the correction for the longitude for 1' error in the altitude. Take from the last column but one of Table II. the nearest or mean bearing, and with this bearing and the given latitude take out the correction as before. This will be the correction for 1' of altitude, which, when the observed altitude is too small, is allowed towards the East or West, according as the body observed is East or West of the meridian, and vice versa when it is too large. Thus for Lat. 40 and Bearing 74 we have 1''35. And for Lat. 40 and Bearing 54 l'-61. In this case as 54 comes between 51 and 57 we take the mean of l-'"67 and l'-55, the corrections given by them. If we wish to correct the H.A. ; we multiply the correction found as above by 4 to obtain seconds of time ; then if the observed altitude is too small it will make the H.A. too great, and vice versa, the correction must therefore be allowed accordingly. Explanation of the Alt.- Azimuth Eule. In a spherical triangle PZS, where PS represents the polar dist., ZS the zenith disk, PZ the co-lat., and PZS the azimuth, we have + Cos. PS -Cos. PZ Cos. ZS Cos. PZS = or Cos. A z = Sin. PZ Sin. ZS Cos. p- Sin. I Cos. z Cos. I Cos. a | + Cos. p sec. Z-Cos. z tan. I \ Sec. = j -Cos. z tan. I + Cos. p sec. I ]- Sec. a Taking the upper or lower sign according as_p is less or greater than 90. Let Cos. z = A, and Cos. jJ^B, then the above becomes | -A tan. lat. + B Sec. lat. [ Sec. alt. Where A tan. lat. is given by Tab. I., andB Sec. lat. by Tab. II., Cos. z and Cos. p are taken from the last column of Tab. II., which column also contains the Nat-Cosine of the Azimuth. , , . , * -, a /5>?5 ?."H.,i ; BY THE SAME AUTHOE, AND PUBLISHED BY J. D. POTTER, 145, Minories, London. s. d. How to Find the Time at Sea in Less than a Minute ; being a new and accurate method, with specially adapted Tables. FOURTH EDITION 2 6 Short Tables and Rules for Finding the Latitude and Longitude ; including also a new and simple Lunar method. SECOND EDITION 3 Time-Altitudes for Expediting the Calculation of Apparent Time, &c. Enabling the Navigator to find Accurate Time at Ship in a feio seconds ... ... ... ... ... ... 4 The Bearings of the Principal Bright Stars of greater declination than 23 N. or 23 S. ; also those of the Moon and Planets when similarly situated. For Latitudes to 60 and Hour Angles Ih. to 9h 3 A Hand-book for Star Double Altitudes, with Directions for Selecting the Stars, and showing how a single observer may take both the Altitudes, &c. ... ... ... 2 6 Nautical Astronomy Made Easy, showing how all the Rules may be worked by a single table on one page ... ... ... 30 Short, accurate, and comprehensive Altitude-Azimuth Tables ; to show the true bearing of the Sun, Moon, and Planets for each degree of Latitude and Altitude from to 75, and declination 30 N. to 30 S. ; also the Approximate Ship Time 3 6 An Elementary Treatise on Plane and Spherical Trigo- nometry, for young Sea Officers. Originally compiled for H. M. S. Britannia, and suitable for Self-Instruction. NINTH EDITION 6 Published by HARBISON d- SONS, 59, Pall Mall, London. Brief and Simple Methods of Finding the Latitude and Longitude by Single and Double Ex-Meridians, Pole-Star, &c. FOURTH EDITION ... ... ... ... .. ... 3 6 Published by GEORGE PHILIP & SON, 31 & 32, Fleet Street, London. " -u-ujjij yy VC 62543 380028 UNIVERSITY OF CALIFORNIA LIBRARY