The apparent time, with the latitude and longitude of the ship, given, to find the apparent altitude of the moon's centre. Turn the longitude into time, (by Table XXI.) and if in west longitude add it to, but in east longitude subtract it from, the apparent time at the ship; the sum or difference will be the apparent time at Greenwich. From this we may deduce the mean time at Greenwich, which is wanted in finding the moon's right ascension and declination. Take the sun's right ascension from the Nautical Almanac for the preceding noon at Greenwich, and add thereto the correction taken from Table XXXI. corresponding to the hours and minutes of the time at Greenwich; the sum will be the sun's right ascension, which, being added to the apparent time at the ship, will give the right ascension of the meridian, rejecting 24 hours when the sum exceeds 24 hours. Take from the Nautical Almanac the moon's right ascension and declination for the time at Greenwich; then the difference between the moon's right ascension and the right ascension of the meridian, will be the moon's distance from the meridian, with which enter Table XXIII., and take out the corresponding logarithm from the column of rising, and add thereto the log. cosine of the latitude of the ship, and the log, cosine of the declination of the moon; the sum (rejecting 20 in the index) will be the logarithm of a natural number, (Table XXVI.) which, being subtracted from the natural cosine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural cosine of their difference when of the same name, will leave the natural sine of the moon's true altitude; from which subtracting the correction corresponding to the altitude in Table XXIX. there will remain the apparent altitude nearly. EXAMPLE. What was the moon's apparent altitude, April 29, 1836, sea account, at 7h 55m 52′ P. M., in latitude 42° 34′ S., longitude 65° 07′ 30′′ W., from Greenwich? April 29, sea account, or by astronomical account. Apparent time at Greenwich... ....April 28 7h 55m 52 4 20 30 ..April 28 12 16 22 Sun's right ascension, April 28d 12h 16m 22a, by Nautical Almanac..... Apparent time at the ship...... Right ascension of the meridian... D's distance from the ineridian .... Corresponding to which, in the column log. rising, is. 2h 25m 11 7 55 52 10 21 03 12 33 27 2 12 24 4.21027 Latitude D's declination.. 42° 34' S. ..... This altitude would be decreased nearly 2', if the true correction of the altitude, corresponding to the D's horizontal parallax, 59, were used, as may be seen in note, at the bottom of the page. * The apparent time is counted from noon to noon, marking the hours from 1 hour to 24 hours. W may remark, that this process of finding the time at Greenwich is unnecessary when you have a chronometer regulated for mean time at Greenwich, because we can immediately obtain the apparen time, by applying the equation of time, taken from the Nautical Almanac, or from Table IV. A., using a different sign from that in the table. When the distance exceeds 12 hours. you must enter Table XXIII. with the difference between that distance and 24 hours. In strictness you ought, instead of this correction, to use the correction of the moon's altitude corresponding to her apparent altitude and horizontal parallax. This is easily found in Table XIX using the D's horizontal parallax and the apparent altitude found by the above process, and subtracting the tabular correction from 59′ 42". Thus, if the D's horizontal parallax is 59', and the D's apparent The apparent time, with the latitude and longitude of the ship, being given, to find the apparent altitude of the centre of a planet. Turn the longitude into time, (by Table XXI. ;) and if west, add it to, but if east longitude, subtract it from, the apparent time at the ship; the sum, or difference, will be the apparent time at Greenwich. From this we may deduce the mean time at Greenwich, which is required in finding the right ascension and declination of the planet.* Take the sun's right ascension from the Nautical Almanac, for the preceding noon at Greenwich, and add thereto the correction taken from Table XXXI., corresponding to the hours and minutes of the time at Greenwich; the sum will be the sun's right ascension, which, being added to the apparent time at the ship, will give the right ascension of the meridian, rejecting 24 hours when the sum exceeds 24 hours. Take from the Nautical Almanac the planet's right ascension and declination for the time at Greenwich; then the difference between the planet's right ascension and the right ascension of the meridian, will be the planet's distance from the meridian; with which enter Table XXIII., and take out the corresponding logarithm, from the column of rising, and add thereto the log. cosine of the latitude of the ship, and the log. cosine of the declination of the planet; the sum (rejecting 20 in the index) will be the logarithm of a natural number, (Table XXVI.) which, being subtracted from the natural cosine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural cosine of their difference when of the same name, will leave the natural sine of the planet's true altitude; to which add the correction of altitude for parallax and refraction, and we shall get the apparent altitude; observing that this correction is found in Table XVII., in the page corresponding to the horizontal parallax of the planet; the difference between the tabular number and 60 being the correction of the planet's altitude for refraction and parallax. EXAMPLE. What was the planet Jupiter's apparent altitude, April 29, 1836, sea account, at 7 55TM 52* P. M., in latitude 42° 34′ S., longitude 65° 7′ 30′′ W. from Greenwich? April 29, sea account, is by astronomical account. Apparent time at Greenwich.... April 28 7h 55m 52′ 4 20 30 .April 28 12 16 22 's right ascension,‡April 28d 12h 16m 22 by Nautical Almanac Apparent time at the ship..... Right ascension of the meridian 's right ascension, in time ... 6 47 08 's distance from the meridian.... 3 33 55 Corresponding to which, in the column of log. rising, is. 4.60733 This is more easily obtained by a chronometer regulated to Greenwich time, as in the preceding example of finding the altitude of the moon. When the distance exceeds 12 hours, you must enter Table XXIII. with the difference between that distance and 24 hours. The sun's right ascension at noon, April 28, is 2h 23 m 158,and the horary motion 98.484, which, for 12h 16m 22, gives, by Table XXXI., 116" == 1′ 56′′ nearly; adding this to 2h 23m 15s, we get the 's right ascension 2h 25m 11. The planet's right ascension and declination are found by inspection in the Nautical Almanac. This correction is found in page 89, Jupiter's parallax being only 1.5. The tabular correction corresponding to the apparent altitude 7° 54′ is 53′ 26"; subtracting this from 60', we get 6' 34", or nearly 7', for the correction arising from the refraction and parallax The apparent time, the latitude and longitude, given, to find the apparent altitude of a fixed star. RULE. Turn the longitude into time, and add it to, or subtract it from, the apparent time at the ship, according as the longitude is west or east; the sum or difference will be the time at Greenwich. The apparent time at Greenwich may also be found by means of a chronometer, as in the preceding example, page 248. Find, in the Nautical Almanac, the sun's right ascension for the noon preceding the time at Greenwich, and add thereto the correction corresponding to the hours and minutes of the time at Greenwich, (using Tables XXX. XXXI. if necessary ;) the sum will be the sun's right ascension, which being added to the apparent time at the ship, will give the right ascension of the meridian, rejecting 24 hours when the sum exceeds 24 hours. Find the star's right ascension and declination in the Nautical Almanac, or by means of Table VIII., as taught in page 217. The difference between the star's right ascension and the right ascension of the meridian, will be the distance of the star from the meridian. Find in the column of rising of Table XXIII. the logarithm corresponding to the star's distance from the meridian, and add thereto the log. cosine of the latitude of the ship, and the log. cosine of the declination of the star; the sum (rejecting 20 in the index) will be the logarithm of a natural number, (Table XXVI.) which being subtracted from the natural cosine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural cosine of their difference when of the same name, will leave the natural sine of the star's true altitude. The refraction being added to the true altitude, will give the apparent altitude. EXAMPLE. What was the apparent altitude of Aldebaran, at Philadelphia, April 12, 1836, sea account, at 5h 57m 18 in the afternoon, apparent time? The star's right ascension and declination are found by inspection in the Nautical Almanac, as below; this being the shortest and most accurate method of finding them. App. time by astronomical account, April 11d 5h 57m 18. 5 0 36 The apparent time must be taken (as usual) one day less than the sea account, and the hour must be reckoned from noon to noon in numerical succession from 1 to 24. It may also be observed that, if the observer be furnished with a chronometer, regulated to mean Greenwich time, this part of the operation may be saved, reducing the mean time to apparent, by applying the equation Table IV. A., or that found in the Nautical Almanac, as in the preceding rules. If the distance from the meridian exceed 12 hours, you must subtract it from 24 hours, before Method of combining several lunar observations together. As a lunar observation is liable to some degree of uncertainty, on account of the imperfections of the instruments, the unavoidable errors of the observations, and the imperfections in the reductions, it will generally be conducive to accuracy to combine together several observations, taken on the same day, or on two or three successive days; and this may be done in the following manner : After working the lunar observation, and finding the mean time of the observation on the meridian of Greenwich, by either of the preceding methods, we must compare this time with the corresponding time of observation, as shown by the chronometer, and the difference will be the error of the chronometer for mean time at Greenwich, as shown by that lunar observation. Other observations, being taken on the same, or on successive days, and computed in the same manner, will also give the errors of the chronometer, corresponding to these observations respectively. The mean of all these errors, being found, will represent very nearly the error of the chronometer, relative to the mean time at Greenwich, and corresponding to that moment of time which results from taking the mean of all the times of observation at Greenwich, for all the lunar observations. Having obtained in this way the error of the chronometer relative to Greenwich time, and knowing its daily rate of loss or gain, we can determine at any moment the mean time at Greenwich, by the chronometer, as it is given by the mean of all these observations. Comparing this mean time with the corresponding mean time ai the same moment at the ship, as found by taking the sun's altitude, or by any othe of the methods explained in pages 208-218, the difference will be the longitude of the ship, resulting from the mean of all these observations. Hence it appears, that, by the mean of the six lurar observations, when the time oy the chronometer was, April 6, 3h 59m 49, it was 2 04' too slow for mean time at Greenwich. We shall now suppose, that, on April 6a 4h 30m 00", by the chronometer, an altitude of the sun was taken, and the mean time at the ship deduced therefrom, April 6 6 24 56, and that it was required to find the longitude of the ship; the chronometer moving uniformly without gain or loss; we shall have Time by the chronometer... ..April 6d 4h 30m 00 Error of the chronometer by the lunar observations.....add Mean time at Greenwich Mean time at the ship..... 2 04 .April 6 4 32 04 The mean of these three observations makes the chronometer too slow for Green wich time 2m 10; and if we suppose the instrument to be well regulated for mean time, and on July 8 4 10m 15 by the chronometer, the mean time at the ship deduced from the sun's altitude, was July 8d 2h 15m 25, we shall have, Time by chronometer... Error by the lunar observations.. Mean time at Greenwich Mean time at the ship.... ... Longitude west of Greenwich .July 80 4h 10m 15" .July 8 4 12 25 1 57 0029° 15 This process may be used for regulating a chronometer when it has accidentally stopped, or has been allowed to run down. For, by comparing the two above examples, supposing them to have been taken by the same chronometer, The first set gives the error April 68 3h 59m 49 equal to +2m 04' + 6a This is, however, an imperfect method of determining the daily gain or loss of the chronometer, on account of the imperfection of the observations; and is only to be used in cases of absolute need. To find the longitude by the eclipses of Jupiter's satellites. The eclipses of the satellites are given in the Nautical Almanac for mean time at Greenwich, and also for sideral time. There are two kinds of these eclipses-an immersion, denoting the instant of the disappearance of the satellite by entering into the shadow of Jupiter, and an emersion, or the instant of the appearance of the satellite in coming from the shadow. The immersions and emersions generally happen when the satellite is at some distance from the body of Jupiter, except near the opposition of Jupiter to the sun, when the satellite approaches to his body. Before the opposition, they happen on the west side of Jupiter, and after the opposition, on the east side. But if an astronomical telescope is used, which reverses the objects, the appearance will be directly the contrary. The configurations, or the positions in which Jupiter's satellites appear at Greenwich, are given, in the Nautical Almanac, every night, when visible. As these eclipses happen almost daily, they afford the most ready means of determining the longitude of places on land, and might also be applied at sea, if the observations could be taken with sufficient accuracy in a ship under sail, which can hardly be done, since the least motion of a telescope which magnifies sufficiently to make these observations, would throw the object out of the field of view. Having regulated your chronometer for mean time at the place of observation, you must then find nearly the mean time at which the eclipse will begin at that place: this may be done as follows:-Find from the Nautical Almanac the mean time of an immersion, or emersion, and apply thereto the longitude turned into time, by adding when in east, but subtracting when in west longitude; the sum or difference will be nearly the mean time when the eclipse is to be observed at the given place. If there be any uncertainty in the longitude of the place of observation, you must begin to look out for the eclipse at an earlier period; and when the eclipse begins, you must note the time by the chronometer, and after applying the correction for the error of the chronometer, if there be any, you will have the mean time of the eclipse at the place of observation; the difference between this and the mean time in the Nautical Almanac, being turned into degrees, will be the longitude from Greenwich. EXAMPLE. Suppose that, on the 21st of August, 1836, sea account, in the longitude of 127° 55′ W., by account, an immersion of the first satellite of Jupiter was observed, at 10h 24m 47 P. M. mean time. Required the longitude. By Nautical Almanac, the time of immersion is,. By observation, August 21, sea account, or by N A.......August 20th 10 24 47 Longitude in time.... ...August 20th 19h 0m 7 8 35 20 which, being turned into degrees, gives 128° 50′ W. for the longitude of the place of |