Reduce the observed altitude of the sun's limb to the true central altitude, by Problem XIV., page 320. Now, add together the sun's true altitude, its polar distance, and the latitude of the place of observation; take half the sum, and call the difference between it and the sun's true altitude the remainder. Then, to the log. co-secant of the polar distance, add the log. secant of the latitude, the log. co-sine of the half sum, the log. sine of the remainder, and the constant logarithm 6.301030: the sum of these five logarithms, abating 20 in the index, will be the log. rising answering to the sun's distance from the meridian; which will be the apparent time at ship or place, if the observation be made in the afternoon; but if in the forenoon, its complement to 24 hours will be the apparent time; the difference between which and the time of observation, per watch, will be the error of the watch, and which will be fast or slow according as the time shown thereby is later or earlier than the apparent time. Remark. In practice, it becomes absolutely necessary to take several altitudes of the sun's limb, and to note the corresponding times per watch; then, the sum of the altitudes, divided by their number, gives the mean altitude, and the sum of the times, so divided, gives the mean time. Example 1. January 1st, 1825, in latitude 40:27' N., and longitude 54:40. W., the following altitudes of the sun's lower limb were observed, the height of the eye above the level of the sea being 20 feet; required the apparent time of observation and the error of the watch? Sun's true central altitude = 13:50:13% 3.39 13:46:34" Constant log. 6. 301030 Sun's north polar distance = . 112.59.32 Log. co-sec.*=0.035949 Lat. of the place of observation Sum = Half sum =. the June 9th, 1825, in latitude 50°40′ N., and longitude 47:56:15 E., following altitudes of the sun's lower limb were observed, the height of the eye above the level of the horizon being 23 feet; required the apparent time of observation, and the error of the watch? The 10 are rejected from the indices of the logarithmic secant and co-secant; and, with the view of facilitating the future operations in this work, the same plan will be pursued in all the computations. 67. 0.7 Sun's north polar distance = Log. co-secant=0.035967 Latitude of the place of observ. = 50.40. 0 Log. secant = 0.198026 Sun's distance from the meridian = 44411 Log. rising = 5.82944.0 Note. Since the log. rising, in Table XXXII., is only computed to five places of decimals, therefore, in taking out the meridian distance of a celestial object from that Table, answering to a given log. rising, the sixth or right-hand figure of such given log. rising, is to be rejected; observing, however, to increase the fifth or preceding figure by unity or 1, when the figure so rejected amounts to 5 or upwards: thus, in the preceding example, where the log. rising is 5.474667, the meridian distance is taken out for 5.47467; and so on of others. For the principles on which the meridian distance of a celestial object is computed, and hence the apparent time, the reader is referred to "The Young Navigator's Guide to the Sidereal and Planetary Parts of Nautical Astronomy," page 156. Remarks. Altitudes for ascertaining the error of a watch ought to be taken by means of an artificial horizon: one produced by pure quicksilver should be preferred, because it shows, at all times, when placed in a proper position, a truly horizontal plane; and, therefore, the angles of altitude taken therein are always as correct as the divisions on the sextant with which those angles are observed; whereas, altitudes taken by means of the sea horizon are generally subject to some degree of uncertainty, owing to its being frequently broken or ill-defined, by atmospherical haze, at the time of observation; though such altitudes are, nevertheless, sufficiently correct for finding the longitude at sea. In taking altitudes by means of an artificial horizon, it is to be observed, that the angle shown by the sextant will be double the altitude of the observed limb of the object; which is to be corrected for index error, if any : then, half the corrected angle will be the observed altitude of the object's limb above the true horizontal plane; to which, if its semi-diameter, refraction, and parallax be applied, the true central altitude of the observed object will be obtained. There is no correction necessary for dip, because the quicksilver shows a truly horizontal plane, as has been before remarked.* The position of a celestial object most favourable for determining the apparent time with the greatest accuracy, is, when it is in the prime vertical; that is, when it bears either due east or due west at the place of observation, or, if it be circumpolar, when it is in that part of its diurnal path which is in contact with an azimuth circle; viz., when the log. sine of its altitude = log. sine of the latitude + radius — log. sine of its declination; because, then, the change of altitude is quickest, and the extreme accuracy of the latitude not very essentially requisite. The nearer a celestial object is to either of these positions, the nearer will the apparent time, deduced from its altitude, be to the truth; as, then, the unavoidable small errors which generally creep into the observations, or a few miles difference in the latitude, will have little or no effect on the apparent time so deduced. Table XLVII. contains the time or distance of a celestial object from the meridian at which its altitude should be observed, in order to determine the apparent time with the greatest accuracy; and Table XLVIII. contains • The direct rules for applying the necessary corrections to altitudes taken on shore by means of an artificial horizon, will be found at the end of the Compendium of Practical Navigation, towards the latter part of this volume. the corresponding altitude most advantageous for observation. But, since those Tables are adapted to the declination of a celestial object when it is of the same name with the latitude of the place of observation, they will not, therefore, indicate either the proper time or the altitude when those elements are of contrary denominations: in this case, since the sun or other celestial object comes to the prime vertical before it rises, and therefore does not bear due east or west while above the horizon, the observation for determining the apparent time from its altitude must be made while the object is near to the horizon; taking care, however, not to take an altitude for that purpose under 3 or 4 degrees, on account of the uncertain manner in which the atmospheric refraction acts upon very small angles of altitude observed adjacent to the horizon.-See explanation to the abovementioned Tables, pages 119 and 120. METHOD II. Of computing the horary Distance of a celestial Object from the Meridian. If the latitude of the place of observation and the declination of the celestial object be of different names, let their sum be taken,-otherwise, their difference, and the meridional zenith distance of the object will be obtained; to which apply its observed zenith distance, by addition and subtraction, and let half the sum and half the difference be taken; then, To the log. secant of the latitude add the log. secant of the declination, the log. sine of the half sum, the log. sine of the half difference, and the constant logarithm 6.301030; the sum of these five logarithms, abating 20 in the index, will be the log. rising of the object's horary distance from the meridian; and if this object be the sun, the apparent time will be known, as in the last method; and, hence, the error of the watch, if necessary. Example 1. January 10th, 1825, in latitude 40:30: N. and longitude 59:2:30 W., the mean of several observed altitudes of the sun's lower limb was 14:31:47", that of the corresponding times, per watch, 3145, and the height of the eye above the level of the horizon 18 feet; required the apparent time of observation, and the error of the watch? Time of observation, per watch, = Greenwich time = 13 145: + 3.56.10 6:57:55: |