If the sun had passed the meridian to the north of the observer, Z would have been 71° 08′ S., and E=91° 18′ S., whose sine 9.99989, added to 9.99390, gives the sine of the latitude 9.99379, corresponding to 80° 20′ S. EXAMPLE IV. Being at sea, in the latitude of 60° 0' N. by account, when the sun was on the equator (or had no declination) at 1h 0m P. M., per watch, his correct central altitude was 28° 53', and at 3h 0 P. M., per watch, the correct central altitude was 20° 42′, required the true latitude. The calculations would have been the same for south latitude, which would be 59° 59′ S. The computation of A and B might have been dispensed with, for when the declination is nothing, B is nothing, and A is equal to half the elapsed time (11) turned into degrees by Table XXI., being, in this example, 15°; in this case, all the logarithms included between the brackets [] may be omitted. In the preceding examples, both altitudes were supposed to be taken at the same place or station; but as that is seldom the case at sea, the necessary correction for any change of place must be made in the following manner : Let the bearing of the sun be observed, by the compass, at the instant of the first observation; take the number of points between that bearing and the ship's course, (corrected for lee-way, if she makes any,) with which, if less than eight, or with what it wants of sixteen points, if more than eight, enter the traverse table, and take out the difference of latitude corresponding to the distance run between the observations. Add this difference of latitude to the first altitude, if the number of points between the sun's bearing and the ship's course be less than eight; but subtract the difference of latitude from the first altitude, if the number of points be more than eight, and that altitude will be reduced to what it would have been if observed at the same place where the second was.* This corrected altitude is to be used with the second observed altitude in finding the latitude by the above rule. The latitude resulting will be that of the ship at the time of taking the second altitude, and must be reduced to noon by means of the log. EXAMPLE V. In a ship, running N. by E. E. per compass, at the rate of nine knots per hour, at 10h 0m A. M., per watch, the sun's correct central altitude was found to be 13° 18', bearing S. 3 E. by compass; and at 1h 40m P. M., per watch, the sun's central altitude was found to be 14° 15'; the latitude by account being 49° 17′ N., and the sun's declination 23° 28' S. Required the true latitude. This is the only correction necessary to make full allowance for the run of the ship; and the inex perienced calculator must take care not to fall into the error of applying a correction to the elapsed time, as is directed in several works of note, particularly in the "Complete Navigator," by Dr. Mackay. This will appear evident by supposing, in the above Example V., that a second observer, with a watch, regulated exactly like that used by the first, was at rest at the place of the second observation. Then, at the first observation, at the same moment of time by both watches, the first observer would find the sun's altitude 13° 18', and the second observer 12° 49'. At the second observation, the times and altitudes would be alike, so that the elapsed time found by both observers would be the same, and the observations would require no correction, except what arises from reducing the altitude from 13° 18 to 12° 49′, because the second observer is supposed to be at rest, and his observation requires no cor ection. The correction to the first altitude. The time elapsed between the observations was 3 40m, and in that time the ship sailed 33 miles upon the course N. by E. E., which makes an angle of 134 points with the sun's bearing at the first observation S. 3 E., the complement of which to 16 points is 24 points. Now, in Table I., the course 24 points, and distance 33", give 29 miles difference of latitude, which must be subtracted from the first altitude 13° 18', because the ship sailed above eight points from the sun; therefore the first altitude corrected will be 12° 49', which must be used in the rest of the work. COL. 1. El.time [P.M.]3h 40m, Cosec. 10.33559 A........ COL. 2. Cosine 9.95704 COL. 3. .Cosec. 10.39988 diff. alts. 0 43...Sine ...Cosec. 10.37308 sum alts. 13 32 Cosine 9.98777 Cosec. 10.63076 B 26° 05′ S. Cosec. 10.35692 8.09718 Secant 10.00003 Cosine 9.95704 [B less than 50, named N. or S. Uklectin.) [Z less than 90°, and N. or S. like bearing of zenith.] E is the sum of B, Z, if of the same name; difference, if of a different name.] Latitude 48 54 N. Sine 9.87716 If the sun had passed the meridian to the north of the observer, Z would have been 75° 01' S., and E=101° 06′ S., whose sine 9.99180, added to 9.99982, gives the sine of the latitude 9.99162 corresponding to 78° 47′ S. EXAMPLE VI. Sailing N. E. § E. by compass, at the rate of nine knots an hour, at 0 31m 40′ P. M., per watch, the altitude of the sun's lower limb was 28° 20′ above the horizon of the sea, the eye being elevated twenty feet above the surface of the water, and the sun's bearing by compass S. by W.; and at 2h 58m 20 P. M., by watch, the altitude of the sun's lower limb was 16° 41' above the horizon, the eye being elevated as before, the latitude by account, at the time of the last observation, 48° 0′ N., and the declination 13° 17′ S. Required the true latitude at taking the last observation. The correction of these altitudes for semidiameter, parallax, and dip, was twelve miles, (additive,) which makes them 28° 32', and 16° 53′. The refraction corresponding to the first was 2 miles, and for the second 3 miles; and, by subtracting these quantities, we have the true central altitudes, 28° 30′, and 16° 50'. Now, the elapsed time between the observations was 2 26m 40, during which the ship sailed twenty-two miles (at nine miles per hour) in the direction of N. E. & E. per compass, the bearing of the sun at the first observation S. by W. being 123 points distant from the ship's course; and as 124 points want 34 of 16 points, we must enter Table I., and find the course 34 points, and distance 22, corresponding to which in the latitude column is 17 miles, which, being subtracted from the first altitude 28° 30', leaves the corrected first altitude 28° 13'; with this, and the second altitude 16° 50', the latitude is found in the following mannèr:— sum alts. 22 314. Cosine 9.96553 Cosec. 10.41670 B 13° 58′ S. Cosec. 10.61732 diff: alts. 5 41....Sine 8.99640 Secant 10.00215 If the sun had passed the meridian to the north of the observer, Z would have Jeen 65 11' S., and E=79' 09′ S., whose sine 9.99217, added to cosine of C 9.97962, gives the sine of the latitude 9.97179, corresponding to 69° 34' S. EXAMPLE VII. [Same as Dr. Brinkley's, in the Nautical Alınanac for 1800.] The latitude by account 6° 30′ N., sun's declination 5° 30′ N., the sun's correct central altitudes 35° 21', and 70° 01', elapsed time between the observations 2h 20in; required the latitude, the sun passing the meridian south of the observer. El.time [P.M.] 220m, Cosec. 10.52186 Cosec. 10.52386 Cosine 9.97962 A....... sum alts. 52 41 diff. alts. 17 20 C...... Cosec. 11.01843 Cosine 9.78263 Cosec. 10.09947 B 5° 46' N....Cosec. 10.99805 Sine 9.47411 Secant 10.02018 i Sine 9.78060 Cosine 9.90170 [Z less than 90°, and N. or S. like bearing of zenith.] Z Sec. 10.00097 [E is the sun of B, Z, if of the same name, difference, if of a different name.] Z 3 50 N. [B less than 90°, named N. or S. like declin.] Cosine 9.90170 E 9 36 N......Sine 9.22211 Lat. 7 38 N......Sine 9.12381 If the sun had passed to the meridian north of the observer, Z would have been 3° 50′ S., and E=1° 56′ N., whose sine 8.52810, added to the cosine of C 9.90170, is 8.42980, which is the sine of the other latitude 1° 32′ N., so that in this example both latitudes are north. SECOND METHOD of finding the latitude by double altitudes of the sun, when the variation of declination is neglected. This method of finding the latitude dépends on a set of tables (marked XXIII., in this collection,) first prepared by Mr. Douwes, containing three logarithms, titled half elapsed time, middle time, and log. rising. The two former are arranged together as far as six hours; the latter is placed at the end of the table, and is extended, in the present edition, as far as twelve hours. The table with the proper title must be entered at the top with the hour, at the side with the minute, and in the column marked at the top with the seconds; the corresponding number will be the sought logarithm, to which must be prefixed the index of the log, under 0' in the same horizontal line. Thus, to the time 3h 52 10 correspond the log. half elapsed time 0.07138, log. middle time 5.22965, and log. rising 4.67274. In general it will be sufficiently exact to take these logarithms to the pearest 10 seconds, particularly when the sun's zenith distance is great; but if the log. to the nearest second is required, it may be found by taking the difference of the tabular logarithms corresponding to the next greater and next less time, and saying, As 10 is to that difference, so are the odd seconds of time to the correction of the first tabular logarithm, additive if increasing, subtractive if decreasing. Thus, if the log. half elapsed time corresponding to 3h 52m 18s were required, the logs. corresponding to 3 52 10 and 3h 52 20′ are 0.07138 and 0.07119, whose difference is 19; then 10: 19 :: 8: 15; this, subtracted from 0.07138, leaves 0.07123, the sought logarithm. By inverting the process, we may find the nearest second corresponding to any given logarithm. We shall now give the rule for calculating the latitude, adapted to double altitudes of the sun. RULE. To the log. secant of the latitude by account (Table XXVII.) add the log. secant of the sun's declination, (Table XXVII.,) rejecting 10 in each index; the sum is to be called the log. ratio. From the natural sine of the greatest altitude (Table XXIV.) subtract the natural sine of the least altitude, (Table XXIV.;) find the logarithm of their difference, (in Table XXVI.,) and place it under the log. ratio. Subtract the time of taking the first observation from the time of taking the second, having previously increased the latter by twelve hours when the observations are on different sides of noon by the watch; take half the remainder, which call half the elapsed time. With half the elapsed time enter Table XXIII., and from the column of half elapsed time take out the logarithm answering thereto, and write it under the log. ratio. Add these three logarithms together, and with their sum enter Table XXIII. in the column of middle time, where, having found the logarithm nearest thereto, take out the time corresponding, and put it under half the elapsed time. The difference between these times will be the time from noon when the greater altitude was .aken. With this time enter Table XXIII., and, from the column of log. rising, take out the logarithm corresponding, from which logarithm subtract the log. ratio; the remainder will be the logarithm of a natural number, which, being found in Table XXVI., and added to the natural sine of the greater altitude, will give the natural cosine of the sun's meridian zenith distance, which may be found in Table XXIV. Hence the latitude may be obtained by the rules of pages 166, 167. NOTES. 1. If this computed latitude should differ considerably from the latitude by account, it will be proper to repeat the operation, using the latitude last found instead of the latitude by account, till the result gives a latitude nearly agreeing with the latitude used in the computation. 2. This method is best suited to situations where the sun's meridian zenith distance is not much less than half the latitude; for in latitudes where the sun approaches near to the zenith, the observations must be taken much nearer to noon; and the preceding rule, instead of approximating, will in some cases give the results of successive operations wider and wider from the truth. To remedy this difficulty, a set of tables was published, by Dr. Brinkley, at the end of the Nautical Almanac for 1799; but the great variety of cases incident to his method, will hinder it from being generally used. Instead of Dr. Brinkley's method, we may generally use the method of arithmetical computation, called Double Position, which will frequently give, in a more simple manner, the required latitude, as will be shown in Example X.; and, in general, it may be observed, that where Douwes's rule does not approximate, the object is most commonly so situated as not to furnish the necessary observations to obtain a correct latitude, whatever method of computation might be used. 3. The operation is the same whether the sun has north or south declination; and also whether the ship is in north or south latitude. When the sun has no declination, the log secant of the latitude (rejecting 10 in the index) will be the log. ratio; and when the latitude by account is nothing, the secant of the declination (rejecting 10 in the index) will be the log. ratio. This rule, as well as the former, is founded on the supposition that the declination is taken for the middle time between the observations, and that it does not vary during the elapsed time, which, however, rarely happens, and a correction ought to be applied to the latitude on this account. But this correction is generally small; and if it is large, the third method must be used; and when the declinations differ very much from each other, we must use the fourth method. correct The index of this logarithm being, as usual, one less than the number of figures contained in the difference of these natural sines; observing, also, that the altitudes to be used are the central altitudes; that is, the observed altitudes corrected for dip, semidiameter, parallax, and refraction. EXAMPLE VIII. [Same as EXAMPLE I., preceding.] Being at sea, in latitude 46° 30′ N. by account, when the sun's declination was 11° 17' N. at 10h 2m in the forenoon, the sun's correct central altitude was 46° 55 and at 11h 27m in the forenoon, his correct central altitude was 54° 9′; required the true latitude, and true time of the day when the greater altitude was taken. Sun is nat. cosine 's zen. dist. 81740...equal to 35° 10′ N. 's declination.... Lat. in ...... 11 17 N. 46 27 N. The latitude 46° 27' (differing only 3′ from the latitude by account) may be assumed as the true latitude. By means of the time of the second observation from noon above found 32m 40′, the error of the watch may be found; for, in the present example, by subtracting 32m 40 from 12", we have the time of the second observation 11h 27m 20; but the time of the watch was 11h 27m 0; therefore the watch was twenty seconds too slow; a small difference would be found in these numbers, if we were to proportion the logarithms of Table XXIII. to seconds. In the same manner, the error of the watch may be found in the following examples.* EXAMPLE IX. [Same as EXAMPLE V., before given.] In this example the latitude by account is 49° 17′ N.; the sun's declination 23° 28′ S. the first altitude corrected, as before, 12° 49′; the second altitude 14° 15'. Required the true latitude. When the middle time is greater than half the elapsed time, both observations are on the same side of the meridian; otherwise, on different sides; whence it is easy to determine whether the greater altitude be observed before or after noon |