same manner as if those elements were of different denominations; and the correction of altitude is to be applied by subtraction to the true altitude of the object, deduced from observation, in order to find its meridional altitude below the pole. Then, with the meridional altitude below the pole, thus found, and the declination, the latitude is to be determined, by Problem V., page 336. The interval, or limits within which the altitude should be observed, is to be determined in the same manner as if the celestial object were near the meridian above the pole. Example 1. June 20th, 1825, at 11:18:30: apparent time, in latitude 71:50. N., by account, and longitude 65: W., the observed altitude of the sun's lower limb was 5:30:50%, and the height of the eye above the level of the sea 20 feet; required the true latitude of the place of observation? Sun's observed reduced to its true central altitude = 5:33:37" Sun's corrected declination = 23:27:36" N. Sun's north polar distance = 66:32:24? Cor. in Table LII., answering to lat. 70: and declin. 23! = = Correction to lat. 71:50 and dec. 23:27:36" = Computed correction = 0"33". 8, Proportional log. = 2.5045 1.2744 Sun's meridian altitude below the pole = 5:17:39" . 66.32.24 Latitude of the place of observation 71:50: 3" N.; which differs but 3 from the truth. In case of a Fixed Star : Find the apparent time of the star's superior transit above the pole, at the given meridian, by Problem XII., page 317; to this time let 12 hours, diminished by half the variation of the sun's right ascension on the given day, be added, and the sum will be the apparent time of the star's inferior transit below the pole. Then, the rest of the operation is to be performed exactly the same as that for the sun in example 1, as above. Example 2. January 1st, 1825, at 11:500: apparent time, in latitude 71:30. N., by account, and longitude 84:9:30" W., the observed altitude of the star Albireo was 9:33, and the height of the eye above the level of the horizon 19 feet; required the true latitude of the place of observation ? Apparent time of star's transit above the pole 0:35:32: Apparent time of the star's transit below the pole Star's distance from the meridian = = 12:33 20: 11.50. O 0:43 20: Observed altitude of the star Albireo = 9:33: 0? Dip of the horizon for 19 feet = Star's apparent altitude = - 4.11 Cor. in Table LII., answering to lat. 70: and declin. 27: = Diff. to 2% of lat.= 3". 4; now, 3". 4 x 90:+120: = Correction to latitude 71:30: and declin. 27:35:54% = Star's meridian altitude below the pole =9: 5:59" . . 62.24. 6 Latitude of the place of observation = 5 from the truth. . 71:30: 5 N.; which differs but Note. From the above examples, the method of finding the latitude by an altitude of the moon, or of a planet, observed near the meridian below the pole, will appear obvious. Remark. The following ingenious problem for determining the latitude, either at sea or on shore, has been communicated to the author by that scientific and enterprising officer, Captain William Fitzwilliam Owen, of His Majesty's ship Eden, who is so highly renowned for his extensive knowledge in every department of science connected with nautical subjects. PROBLEM. Given the Latitude by Account, the true Altitude of the Sun's Centre, and the apparent Time; to find the true Latitude of the Place of Observation. RULE. Find the mean between the estimated meridian altitude, and the altitude deduced from observation, which call the middle altitude; then, To the log. rising of the apparent time from noon, add the log. co-sine of the latitude, the log. co-sine of the corrected declination, the log. secant less radius of the middle altitude, and the constant logarithm 7.536274 ;* the sum of these five logarithms, abating 30 in the index, will be the logarithm of a natural number, which is to be esteemed as minutes, and which, being added to the sun's true central altitude, will give his correct meridional altitude; and, hence, the true latitude of the place of observation? Example 1. December 22d, 1825, in latitude 8:0. south, by account, at 23:41 15: apparent time, the true altitude of the sun's centre was 74:16; required the true latitude? * This is the log. secant of one minute, with a modified index. Note. By this method of computation, an error of one degree in the latitude by account, in places within the tropics, will produce little or no effect on the latitude resulting from calculation: thus, if the latitude by account be assumed at 7:0, or at 9:0, the resulting latitude, or that deduced from computation, will not differ more than one minute from the truth; and the same result would be obtained, if the altitude were observed at the distance of an hour from noon : provided, always, that the measure of the interval from noon be very correctly known. Example 2. December 23d, 1825, in latitude 50:0: N., by account, at 1:14:15: apparent time, the altitude of the sun's centre was 13:58; required the true latitude? Co-latitude of the place of observation=39:14:north; hence the true lati tude is 50:46 north, which is 2. less than the result by spherical trigonometry: the correct latitude being 50:48 north. If the latitude by account be assumed at 51:48, the latitude by computation will be 50:50; being, in this instance, only two minutes morc than the truth. Note. The above method of finding the latitude is, as far as I am aware, perfectly original; it is exceedingly well arranged, and it affords a direct and general solution to the problem given, for the same purpose, in page 354 the apparent time, or the measure of the interval from : noon, must, however, be very correctly known; although, in places distant froni the equator, or where the sun does not come very near to the zenith of the place of observation, an error of a few minutes in the time will not very materially affect the latitude: thus, in the last example, an error of two minutes in the interval from noon would only produce an error of six minutes in the latitude; and in the first example, where the sun passes nearer to the zenith, it would produce an error of eight minutes in the latitude. As this method does not labour under any restraint, or since it does not require that the interval from noon should be governed by the object's meridional zenith distance, the observation may therefore be taken at any hour before or after the sun's transit; and this is a peculiarity that gives it a most decided advantage over the method contained in the abovementioned page. PROBLEM XIII. Given the Longitude of a Place, the Sun's Declination and Semi-diameter, and the Interval of Time between the Instants of his Limbs being in the Horizon; to find the Latitude of that Place. RULE. Reduce the apparent time, per watch, of the rising or setting of the sun's centre to the corresponding time at Greenwich, by Problem III., page 297; to which time let the sun's declination be reduced, by Problem V., page 298. To the logarithm of the sun's semi-diameter, reduced to seconds, add the arithmetical complement of the logarithm of the interval of time, expressed in seconds, between the instants of the sun's limbs being in the horizon, and the constant logarithm 9. 124939; the sum of these three logarithms, rejecting 10 in the index, will be the logarithmic co-sine of an arch. Now, |