Cosine.. 9.87241 corresponding to which, in the column P. M., is 5h 34m 25', the apparent time of day, which agrees nearly with the first method. By the preceding method you may find the beginning or ending of the twilight, by calculating the hour when the sun's zenith distance is 108°, (or when the sun is 18° below the horizon;) for by observation it has been found that the twilight begins or ends when the sun is at that distance from the zenith. EXAMPLE III. Required the time of beginning and ending of the twilight, in the latitude of 42° 23′ N., when the declination is 23° 27' N. Zenith distance Co-latitude. ...... Polar distance Half-sum..... cosine 9.43480 which corresponds to 2h 6m 20a A. M., and 9h 53m 40 P. M. Therefore, the first appearance of the twilight in the morning was at 2b 6m 20°, and the end of it in the evening at 9h 53 40", apparent time. THIRD METHOD. If the sun's declination, and the latitude, be both north or both south, take their difference, but if one be north and the other south, take their sum, and from the natural cosine of this difference, or sum, subtract the natural sine of the true altitude, (both being found in Table XXIV.;) find the log. of their difference, (in Table XXVI.) add thereto the log. secant of the latitude (from Table XXVII.) and the log. secant of the sun's declination, (from the same table,) rejecting 10 in each index; the sum of these three logarithms being found in the column of rising (Table XXIII.) the hours, minutes, and seconds, corresponding, will be the apparent time from noon; and by applying the equation of time to the apparent time, we get the mean time. The preceding examples I. and II. are thus worked by this third method: sponding to which, in column rising, is apparent time 5h 34m 300 Equation of time... Mean time of observation. Time per watch.. Watch too slow.... ...sub. 3 49 5 30 41 41 agreeing nearly with the other methods. The differences between the results of the different methods arise chiefly from not taking notice of the seconds in the angles, and sometimes from not having the natural sines and cosines to 6 or 7 places of decimals; and we remark generally, that it is always best to retain the seconds in the calculation. This is easily done, in the first and second methods, by means of the columns A, B, of proportional parts in Table XXVII.; and by retaining the seconds, we are sure to obtain a more correct result in the calculation. TO FIND THE TIME AT SEA BY THE MOON'S ALTITUDE. HAVING a chronometer which is pretty well regulated to Greenwich time, we can use the moon's altitude for finding the mean solar time at the ship, which is required in determining the longitude. For, in the present improved state of the Nautical Almanac, we can easily find the moon's right ascension and declination for that time at Greenwich, without the very troublesome operation of interpolating for the second and third differences, as was necessary in the former arrangement of that ephemeris. Even without a chronometer thus regulated, the time at Greenwich can be obtained, if we know the longitude of the ship, as well as the mean time at the place of observation, by a watch that will give it with a considerable degree of accuracy; because, by adding the longitude in time to the time by the watch, if the longitude be west, or subtracting it, if the longitude be east, we shall obtain the corresponding time at Greenwich. We must, however, always keep in mind, that the accuracy of an observation of this kind depends on the certainty with which the time at Greenwich is computed; because an error in this estimate affects the moon's right ascension and declination, which frequently vary rapidly, as may be seen by the inspection of the Nautical Almanac, where we shall find that in a minute of time the right ascension may vary more than 2, and the declination more than 15". When we wish to ascertain the time by this method, we must observe, with a fore observation, the altitude of the moon's round limb, and at the same instant the time by the watch or chronometer, which is supposed to be regulated to Greenwich time. With this time at Greenwich we must take from the Nautical Almanac the sun's right ascension, the moon's right ascension, the moon's declination, the moon's horizontal parallax, and the moon's semi·liameter, to which we must add the augmentation from Table XV. To the observed altitude we must apply the correction of the moon's semidiameter, by adding it, if the lower limb be observed, or subtracting it, if the upper limb be observed; from this sum or difference we must subtract the dip of the horizon, and we shall obtain the moon's central altitude. To this we must add the correction for parallax and refraction, and we shall obtain the moon's correct altitude, which is to be used in the rest of the calculation. This correction for parallax and refraction can easily be found, as in page 171, by means of Table XIX., by subtracting the tabular number, corresponding to the altitude and horizontal parallax, from 59′ 42′′; the remainder will be the correction for parallax and refraction, to be used as above; and then we find the time by the following rule: RULE. Add together the moon's correct altitude, the ship's latitude, and the polar distance; from the half-sum subtract the moon's correct altitude, and note the remainder; then add together the log. secant of the latitude, the log. cosecant of the polar distance, (rejecting 10 from each index,) the log. cosine of the half-sum, and the log. sine of the remainder; half the sum of these four logarithms will be the log. sine of half the hour angle; take out the corresponding time in the column marked P. M., in Table XXVII., and apply it to the moon's right ascension, by subtracting when the moon is east of the meridian, or adding when west of the meridian; the sum or difference will be the right ascension of the meridian. From the right ascension of the meridian (increased by 24 hours if necessary) subtract the sun's right ascension; the remainder will be the apparent time at the ship, and by applying to it the equation of time found in the Nautical Almanac, we shall get the required mean solar time at the meridian of the place of observation. EXAMPLE. When the mean time at Greenwich, by the chronometer, was, Nov. 29d, 2h 52m astronomical account, the altitude of the moon's upper limb was observed, when west of the meridian, and found to be 60° 25′ 8′′, the latitude of the place 30° 20′ N, and the dip 4'. Required the mean time of observation. We have from the Nautical Almanac, for the time, Nov. 29d 2h 52m, the sun's right ascension, 16h 22m 453; the moon's right ascension, 9h 26m 23; the moon's declination, 20° 32′ 47′′ N., or polar distance, 69° 27' 13"; the moon's horizontal parallax, 54′ 23′′; the moon's semidiameter, 14' 49" + Aug. Table XV. 14" 15′ 3′′. D's observed altitude, upper limb.......,. Dip....... D's central altitude... Parallax and refraction 59′ 42′′ . Cor. Tab. XIX. 33′ 08′′ ....add D's correct altitude..... 60 06 05 26 34 60 32 39 Sum (being west of the merid.) gives right ascension of the meridian 11 29 59 Subtract the sun's right ascension 2h 03m 360 9 26 23 16 22 45 It very frequently happens, that, a few minutes before or after taking the sun's meridian altitude for the determination of the latitude, we can observe the moon's altitude for the regulation of the time; and as the latitude by observation is then known accurately, without depending on the ship's run for any considerable length of time, it will operate to render the regulation of the chronometer by the moon's altitude more accurate. In like manner, if we observe the latitude by the moon's meridian altitude, we can, at nearly the same time, take an observation of the sun's altitude to regulate the chronometer. TO FIND THE TIME AT SEA BY A We may use either of the large planets, Jupiter, Saturn, Mars, or Venus, for determining the time at sea; and the process is very nearly the same as that in the preceding section, where the moon's altitude is used. In this case, we must ascertain the time of observation, reduced to the meridian of Greenwich, either by a chronometer regulated to that meridian, or by knowing pretty nearly the mean time of observation at the ship, and the longitude; for by adding the longitude in time to the mean time at the ship by the watch, if the longitude be west, or subtracting the longitude if it be east, we shall obtain the corresponding time at Greenwich. With this time at Greenwich, we must take, from the Nautical Almanac, the sun's right ascension, the planet's right ascension, the planet's declination, or polar distance. The parallax and semidiameter might also be noticed, but the corrections from these quantities are so small that they may be neglected, as only amounting to a few seconds. Then from the observed central altitude of the planet we must subtract the dip and the refraction, and we shall obtain the planet's correct altitude. With these we may find the time by the following rule: RULE. Add together the planet's correct altitude, the ship's latitude, and the polar distance; from the half sum subtract the planet's correct altitude, and note the remainder; then add together the log. secant of the latitude, the log. cosecant of the polar distance, (rejecting 10 from each index,) the log. cosine of the half-sum, and the log. sine of the remainder; half the sum of these four logarithms will be the log. sine of half the bour angle; take out the corresponding time in the column marked P. M., Table XXVII., and apply it to the planet's right ascension, by subtracting from the right ascension when the planet is east of the meridian, or adding when west of the meridian; the sum or difference will be the right ascension of the meridian. From the right ascension of the meridian (increased by 24 hours if necessary) subtract the sun's right ascension; the remainder will be the apparent time at the ship; and by applying to it the equation of time, found in the Nautical Almanac, we shall get the required mean solar time, at the meridian of the place of observation. EXAMPLE I. In the latitude 42° 22′ N., and longitude 70° 15′ W., on May 26d 7h 18m 35', astronomical time, by a watch which was very nearly regulated for mean time at the ship, observed the central altitude of the planet Jupiter, by a fore observation, and found it to be 32° 16′ 23′′; the planet being to the west of the meridian, and the dip 4' 8". Required the mean time of observation at the ship. Adding the longitude 4h 41m to the time by the watch, we get the mean time at Greenwich, May 26 11h 59m 35; and with this time we get, from the Nautical Almanac, the sun's right ascension 4h 15m 04; Jupiter's right ascension 7h 8m 32'; Jupiter's declination 22° 48′ 39′′′ N., or polar distance 67° 11" 21". When very great accuracy is required, we may notice the parallax in altitude, which is found in Table X. A., and is to be added to the correct altitude computed by the above rule. We may also find the correction of refraction and parallax, by entering Table XVII. in the page corresponding to the horizontal parallax of the planet, and taking out the corresponding number, which, being subtracted from 60, gives the correction for parallax and refraction, at one operation. |