subtraction to the true altitude of the object, its meridional altitude below the pole will be obtained. With the meridional altitude below the pole, thus found, and the reduced declination of the object, the latitude is to be determined by Problem V., page 392. Note. The interval between the time of observation and the moment of transit (viz., the 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.-See the first paragraph, to Problem IX., in page 414. Example. At sea, June 21st, 1836, at 12:57 15 mean time, in latitude 71:55 north, by account, and longitude 65.0 west, the mean of several observed altitudes of the sun's lower limb, reduced to the true central altitude, was 5:51:30 south of the observer; required the true iatitude of the place of observation? 0:28:52 Prop. log. 0.7948 5.51.30 South. Appar. time from midn. 0:55:43: Twice its prop. log. Correction of altitude. True central altitude of the sun Sun's meridional altitude below the pole 5:22:38" South. Latitude of the place of observation 71:55: 3 North. Note. If the object be a fixed star, let the mean time of its superior transit above the pole at the given meridian, be determined by Problem VIII., page 348; to this time let 12 hours diminished by half the di urnal increase of the mean sun's right ascension, be added (viz., 12-1:58:28=11:58 1:72), and the sum, abating 24 hours, if necessary, will be the mean 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 the Example, page 422., Remark. The following ingenious problem for determining the latitude, was communicated to the author by that scientifical 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. cosine 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. December 22nd, 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. co-secant of one minute with a modified index. Sun's corrected declination = 23. 27. 0 S. Log. co-sine = 9.962562 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. But I have to observe, that this method will not answer in places to the northward of the Tropic of Cancer, or to the southward of the Tropic of Capricorn. PROBLEM XIII. Given the Longitude of a Place, the Sun's Declination and Semidiameter, 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 mean time, per watch, of the rising or setting of the sun's centre to the corresponding time at Greenwich, by Problem III., page 342; to which time let the sun's declination be reduced, by Problem XIV., page 357. To 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, add the logarithm of the sun's semidiameter, reduced to seconds, 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, to the logarithmic sine of the sum of this arch and the sun's reduced declination, add the logarithmic sine of their difference; then, half the sum of these two logarithms will be the logarithmic sine of the latitude of the place of observation. Example. At sea, July 13th, 1836, in north latitude, and longitude 120° west, the sun's lower limb, at the time of its setting, was observed to touch the horizon at 8:524: mean time, and the upper limb at 8:9:30: :; required the latitude of the place of observation? Mean time of sun's centre setting=8:524:+89 30:+2 = 8:7"27! Longitude 120: west, in time Mean time at Greenwich= 8.0. 0 16:727: -The sun's declination reduced to this time is 21:42:57" north. Constant logarithm Arch 7.609065 . 2.975616 Lat. of the place of observation= 50:47:58 Log. sine Remark. In this method of finding the latitude, it is indispensably necessary that the interval of time (per watch) between the instauts of the sun's lower and upper limbs touching the horizon be determined to the nearest second; otherwise the latitude resulting therefrom may be subject to a considerable error, particularly in places where the limbs of that object rise or set in a vertical position; which is frequently the case in parts within the tropics. *This is the ar. comp. of the prop. log. of 24 hours, esteemed as minutes. SOLUTION OF PROBLEMS RELATIVE TO MEAN TIME, &c. Time, as inferred directly from observations of the heavenly bodies, is denominated either apparent or mean solar time. Apparent time is that which is deduced from an altitude of the sun; and mean time from the altitudes of the moon, stars, or planets. Mean time arises from a supposed uniform motion of the sun: hence, a mean solar day is always of the same determinate length; but the measure of an apparent day is ever variable,-being longer at one time of the year, and shorter at another, than a mean day; the instant of apparent noon will, therefore, sometimes precede, and at. other times follow, that of mean noon. The difference of those instants is called the equation of time; which equation is expressed by the difference between the sun's true right ascension and his mean longitude, corrected by the equation of the equinoxes in right ascension, and converted into time at the rate of 1 minute to every 15 minutes of motion, &c. &c. The equation of time is always equal to the difference between the times shown by an uniform or equable-going clock, and a true sun-dial. The sun's motion in the ecliptic is constantly varying, and so is his motion in right ascension; but since the latter is rendered further unequal, on account of the obliquity of the ecliptic to the equator, it hence follows that the intervals of the sun's return to the same meridian become unequal, and that he will gradually come to the meridian of the same place too late, or too early, every day, for an uniform motion, such as that shown by an equable-going watch or clock. It is this retardation or acceleration of the sun's coming to the meridian of the same place, that is called the equation of time; which implies a correction that is additive to, or subtractive from, the apparent time, in order to reduce it to equable or mean time; and vice versa, which is subtractive from, or additive to, the mean time in order to reduce it to apparent time. But, as this essentially useful element has been particularly treated of in the Explanatory Articles between pages 309 and 316; the reader is, therefore, requested to refer to those pages, where he will find all the information that he can desire relative to the equation of time. The young navigator will please to bear in mind, that the equation of time relating to nautical operations (like the sun's declination), is that which is contained in page II. of the month in the Nautical |