In this manner I have often obtained the altitudes in much less time than they could have been obtained by other calculations. The same method may be used for finding the sun's altitude, when taking an azimuth, by noting the times of taking the observations by a watch, and taking two altitudes, the one before, the other after the observation, and proportioning the altitudes as above. Any person who wishes to calculate strictly the apparent altitudes, may proceed according to the following rules. The apparent time, the ship's latitude and longitude, and the sun's declination given, to find the apparent altitude of his centre. RULE. With the apparent time from noon, enter Table XXIII. and from the column of rising take out the logarithm corresponding, to which add the log. co-sine of the latitude, and the log. co-sine of the sun's declination; their sum, rejecting 20 in the index, will be the logarithm of a natural number, which being subtracted from the natural co-sine of the sum of the declination and latitude, when they are of different names, or the natural co-sine of their difference when of the same name, will leave the natural sine of the sun's true altitude at the given time. The refraction less parallax being added to the true altitude, will give the apparent altitude. In general it will be near enough to take out the refraction only from Table XII. and neglect the parallax. EXAMPLES. Required the true altitude of the sun's centre, in lat. 49° 57' N. and long. 75° W. July 26, 1820, at 6h. 56m. 30s. in the morning, sea account? H. M. S. 12 อ 0 App. time 6 56 SO H. M. S. App. time from noon 3 21 SO Latitude Declin. at that time App. alt. EXAMPLE II. What will be the true altitude of the sun's centre in the latitude of 39° 20′ N. and the longitude of 40° 50′ W. November 26, 1820, at 3h. 21m. 30s. apparent time in the afternoon, sea account? log. co-sine . 4.55900 9.88844 9.97054 The apparent time, the latitude and longitude given, to find the apparent altitude of a fixed star. Turn the longitude into time, and add it to, or subtract it from, the apparent time* The apparent time must be taken (as usual) one day less than the sea account, and the hours 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, with a different sign from that in the Table, as is taught in the introduction to the tables. at the ship, according as the longitude is west or east, the sum or difference will be the time at Greenwich. Find, in the Nautical Almanac, the sun's right ascension for the noon preceding the time 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. Find the star's right ascension and declination in Table VIII. for the year 1820, and correct them for the years elapsed since that time, by means of the annual variations given in the same table, and you will obtain the star's right ascension and declination at the time of observation.* 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. co-sine of the latitude of the ship, and the log. co-sine of the declination of the star, the sum, rejecting 20 in the index, will be the logarithm of a natural number (Table XXVI.) which subtracted from the natural co-sine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural co-sine 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, 1820, sea account, at 5h. 57m. 18s. in the afternoon, apparent time? In Table VIII. the right ascension of Aldebaran for 1820, is 4h. 25m. 36s. and the variation for 3 months is 1" to be added because the time is after 1820; hence the right ascension at the given time 4h. 25m. 37s. The declination of the star for 1820 is 16° 8' N. its variation for 3 months being neglected. * If any of the 24 bright stars are used, whose right ascension or north polar distances are given for every 10 days in the year in the Nautical Almanac, we can obtain from it by inspection the right ascension, and deduce the declination from the polar distance, and the numbers thus found having been corrected for aberration and nutation, are to be used as rather more accurate. If the distance from the meridian exceed 12 hours, you must subtract from 24 hours before entering table XXIII. The apparent time, and 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 time at Greenwich. Take the sun's right ascension out of the N. A. 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 N. A. the moon's right ascension and declination for the time immediately preceding and following the time at Greenwich, and proportion for the time at Greenwich, by means of Table XXX. and you will obtain the moon's right ascension and declination at the time of observation. Turn the moon's right ascension into time (by Table XXI.) and the difference be tween that time and the right ascension of the meridian, will be the moon's distancef from the meridian; with which enter table XXIII. and take out the corresponding logarithm from the column of rising, and add thereto the log. co-sine of the latitude of the ship, and the log. co-sine of the declination of the moon, the sum, rejecting 20 in the index, will be the logarithm of a natural number (Table XXVI.) which subtracted from the natural co-sine (Table XXIV.) of the sum of the declination and latitude when of different names, or the natural co-sine 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, 1820, sea account, at 7h. 55m. 52s. P. M. in lat. 42° 34' S. long. 65° 7′ 30′′ W. from Greenwich? The apparent time is to be counted from noon to noon, as directed in the preceding note. + When the distance exceeds 12h. 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, cor responding to her apparent altitude and horizontal parallax. To find the Longitude by the Eclipses of Jupiter's Satellites. The eclipses of the satellites are given in page III. of the month of the Nautical Almanac for mean time at Greenwich. 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 near 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 laid down every night, when visible, in page XII. of the month of the Nautical Almanac. 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 objects out of the field of view. As these eclipses are given in the Nautical Almanac in mean time, it is necessary to regulate your watch to mean time;* this is easily obtained from the apparent time by applying to the latter the equation of time taken from the Nautical Almanac, by adding or subtracting according to the directions in the column from whence the equation was taken; hence the error of a watch with respect to mean time may be ascertained. The watch being thus regulated, you must then find nearly the time at which the eclipse will begin at the place of observation; this may be done as follows: Find from the Nautical Almanac the 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 watch, and after applying the correction for the error of the watch, if there be any, you will have the mean time of the eclipse at the place of observation; the difference between this and the time in the Nautical Almanar, being turned into degrees, will be the longitude from Greenwich. EXAMPLE. Suppose that on the 21st of August, 1820, sea account, in the longitude of 127° 55′ W. by account, an immersion of the first Satellite of Jupiter was observed at 7h. 12m. 328. P. M. mean time. Required the longitude? By N. A. immersion By observ. Aug. 21, sea account, or by N. A. Longitude in time..... Aug. 20th. 15h. 47′ 52" 8 35 20 which turned into degrees gives 128° 50′ W. for the longitude of the place of obser vation. To find the Longitude by Eclipse of the Moon.. The determination of the longitude by an eclipse of the moon is performed by comparing the times of the beginning or ending of the eclipse, as also the times when any number of digits are eclipsed, or when the earth's shadow begins to touch or leave any remarkable spot on the moon's face; the difference of time between the like observations made at different places, turned into degrees, will be the difference of longitude of those places. When the beginning or end of an eclipse of the moon is observed at any place, the longitude of that place may be easily found by comparing the time In the Almanacs published before 1805, the apparent time of the eclipses was given instead of the mean time. of observation with the time given in the Nautical Almanac, for the differ ence between the observed time of beginning or ending, and the time given in the Nautical Almanac, will be the ship's longitude in time, which may be turned into degrees by Table XXI. Thus if the beginning of an eclipse of the moon was observed March 30, 1820, sea account, at 9h. 59 m. the time by the N. A. being March 29 or March 30, sea account, at 5h. 16§m. their difference 4h. 43m. is the longitude of the place of observation=700 45', which is east from Greenwich, because the time at the place of observation is greatest. To find the Longitude by a perfect time-keeper or chronometer. It was before observed, that if a chronometer could be made in so perfect a manner as to move uniformly in all places, and at all seasons, the longitude might easily be deduced therefrom, by comparing the time shown by the chronometer, regulated to the meridian of Greenwich (or some other known meridian) with the mean time at the place of observation. For the difference of these times would be the difference of longitude between that meridian and the place of observation. The moderate price of good chronometers now, in comparison with their values many years since, together with the various improvements in their construction, have caused this method of determining the longitude to be much more used within a few years, than it was when the first editions of this work were published: we shall therefore explain more fully the use of this instrument, and the methods of regulation. at sea. If a chronometer is to be used on a voyage, it must be adjusted, and its rate of going ascertained, before sailing. This may be done by taking altitudes of the sun or some other heavenly body, and finding therefrom the apparent time of observation, by any of the methods before given in pages 154-161. To this time must be applied the equation of time, found in page II. of the month of the Nautical Almanac, or in Table IV. A (reduced to the moment of observation by means of Table VI. A) by adding the equation to, or subtracting it from, the apparent time, according to the directions given in or at the top of each column of the table, the sum or difference will be the mean time of observation, being the same time as would be shown by a chronometer whose motion is perfectly uniform. Comparing this mean time of observation with the time by the chronometer, shows how much it is then too fast or too slow for the meridian of the place of observation; and by repeating the operation on a future day, the rate of going may be ascertained. If it is found to gain or lose a few seconds or parts of a second per day, that allowance must be made on all future observations Thus, if on the 1st of June, 1824, at 5h. 10m. 20s. by the chronometer, the mean time, deduced from an observation of the sun's altitude, was 5h. 12m. 40s. the chronometer would then be too slow by the difference of those times 2m. 20s. and if on the 21st of June following the time by the chronometer was 4h. 15m. 35s. when the mean time was 4h. 18m. 17s. the chronometer would then be too slow by the difference of those times or 2m. 42s. and the rate would have varied in 20 days from 2m. 20s. to 2m. 42s. which is a difference of 22 seconds in 20 days, being 17 seconds per day, and this rate must be allowed on all future observations at sea, until a new regulation can be obtained, at some place whose longitude is known. It is best to have a considerable number of days interval between the two observations for fixing the rate, by which means it may be determined to tenths of a second, the absolute error of the observations being reduced in finding the daily rate, by dividing by the number of days. Thus if the above difference of 22 seconds had been erroneous 2s. and the true value 20s. the daily rate would be one second instead of 1s.1, varying only one tenth of a second, notwithstanding the observations on which the rate was established, contained an error of 2 seconds. The Chronometers most celebrate. for correctness are those made by Mr. French, London, and for sale by JAMES LADD, No. 30 Wall-street, New-York, who mechanically understands that valuable instrument. Proprietor. |