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 at the same moment at the ship, as found by taking the sun's altitude, or by any other 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 lunar observations, when the time by the chronometer was, April 64, 3h 59m 49, it was 2m 04 too, slow for mean time at Greenwich.

We shall now suppose, that, on April 6 4h 30m 00s, by the chronometer, an altitude of the sun was taken, and the mean time at the ship deduced therefrom, April 6 6h 24m 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

2.04

.April 6 4 32 04
.April 6 6 24 56

1 52 5228° 13

The mean of these three observations makes the chronometer too slow for Greenwich time 2m 10; and if we suppose the instrument to be well regulated for mean time, and on July 8d 4h 10m 15 by the chronometer, the mean time at the ship, deduced from the sun's altitude, was July 84 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 8d 4h 10m 15′
.add 2 10

,July 8 4 12 25

.July 8 2 15 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 6d 3h 59m 49s equal to +2m 04'
The second set gives the error July 7 4 16 11 equal to +2 10

Gain in 92 days

+ 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 appear-ance 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.. Longitude in time......

...August 20th 19h 0m 7 ..August 20th 10 24 47

8 35 20

which, being turned into degrees, gives 128° 50′ W. for the longitude of the place of

To find the longitude by an 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 in the moon's face; the difference of these times 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 of observation with the time given in the Nautical Almanac ; for the difference between the observed mean time of beginning or ending, and the mean 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 October 25, 1836, sea account, at 5b 21", mean time; the mean time at Greenwich by the Nautical Almanac being October 24, or October 25, sea account, at Oh 38m, their difference, 4h 43m, is the longitude of the place of observation=70° 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 mean 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 prices 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 very much used within a few years; we shall therefore explain fully the use of this instrument, the methods of regulating and ascertaining its rate of going, and give examples of the calculations for finding the longitude.

If a chronometer is to be used on a voyage, it must be adjusted, and its rate of going ascertained, before sailing. This is most conveniently done on shore by observing, with a transit instrument, the times of the transits of the sun, or some fixed star, over the meridian, as is taught in pages 221-224. If you have no instrument of this kind, the regulation may be made by taking altitudes of the sun or some other heavenly body, and finding therefrom the mean time of observation, by any of the methods before given in pages 208-218. The best way of making these last observations on land, is by an artificial horizon of quicksilver; finding and correcting the altitudes in exactly the same way as in computing the latitude in page 204. Comparing the mean time of observation, obtained in this way, with the time by the chronometer, shows how much it is then too fast or too slow for the meridian of the place of obser vation; 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 at sea. Thus if, on the 1st of June, 1836, at 5h 10m 20, by the chronometer, the mean time, deduced from an observation of the sun's altitude, was 5h 12m 40, the chronometer would then be too slow by the difference of those times, 2m 20; and if, on the 21st of June following, the time by the chronometer was 4h 15m 35, when the mean time was 4h 18m 17, the chronometer would then be too slow by the difference of those times, or 2m 42; and the rate would have varied, in 20 days, from 2m 20s, to 2m 42s, which is a difference of 22* in 20 days, being 1'.1 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, since by this 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 had been erroneous 2, and the true value 20, the daily rate would be one second, instead of 1.1, varying only one tenth of a second, notwithstanding the observations on which the rate was established contained an error of two seconds. Having regulated a chronometer, in the manner first mentioned, at a place whose longitude from Greenwich is known, it is easy to find how much it is too fast or too

*See Tab. LVII.

slow for the meridian of Greenwich, by reducing the mean time at the place of the observer, as found by observations, to the meridian of Greenwich, by adding the longitude if west, subtracting if cast; the sum or difference will be the mean time of observation in the meridian of Greenwich; the difference between this and the time given by the chronometer, shows how much it is too fast or too slow for Greenwich mean time. Thus, by adding the longitude, which we shall suppose to be 4 56TM, to the mean time of the above observation, 5h 12m 40, we get 105 8m 40s for the mean time at Greenwich; from which subtracting the time by the chronometer, 5h 10m 203, we obtain 4h 58m 20 for the error of the chronometer relative to mean time at Greenwich; being too slow for that time.

The chronometer having been thus regulated to Greenwich time, and the daily rate of its going ascertained, if this rate should remain unaltered, the time at Greenwich will be known by it, at any moment at sea; and if at that moment, by any observation of the sun, moon, planet, or a fixed star, the mean time at the ship be found by any of the methods explained in pages 208, &c., the difference between this mean time at the ship, and the mean time at Greenwich, shown by the chronometer, will be the longitude, which may be turned into degrees and minutes by Table XXI.

EXAMPLE I.

Wishing to regulate a chronometer, in a place whose latitude is 51° 30′ N., and longitude 130° E. from Greenwich, I observed, October 10, 1836, at 8h 21m A. M., sea account, by a chronometer, the altitude of the sun's lower limb, by a fore observation, 13° 32, the correction for semidiameter, parallax, and dip, being 12. It is required to find the error of the chronometer for mean time at Greenwich.

The mean time of this observation, at the meridian of the ship, computed as in Example I., page 209, is 7h 54m 17 A. M., or October 9d 19h 54m 17, astronomical account. From this subtract* the longitude 130°, turned into time 8h 40m, (by Table XXI.) we get the corresponding mean time at Greenwich, Oct. 94, 11h 14m 17; and as the time by the chronometer is, October 9a, 20h 21m 00s, it is too fast for mean time at Greenwich by the difference of those two quantities, or 9h 6m 43a.

EXAMPLE II.

May 10, 1836, at 5h 30m P. M., sea account, by a chronometer, in latitude 39° 54′ N, in a place whose longitude was known to be 35° 45′ E. from Greenwich, the altitude of the sun's lower limb by a fore observation was 15° 45', the correction for dip, parallax, and semidiameter, being 12. It is required to find the error of the chronometer for mean time at Greenwich.

The mean time of this observation, computed as in Example II., page 210, is May 9 5h 30m 39, astronomical computation. From this subtract the longitude, 35° 45', turned into time, 2h 23m, by Table XXI.; the remainder, May 9 3 7 39', is the mean time at Greenwich. The difference between this and the time by the chronometer, 5h 30m, is 2h 22m 21', which expresses how much the chronometer is too fast for Greenwich mean time.

EXAMPLE III.

Suppose that, on July 27, 1836, sea account, the mean time was found, by an altitude of the sun, to be 1h 11m 16 P. M., when, by a chronometer well regulated to mean time at Greenwich, it was 4h 3m 8 P. M. Required the longitude.

Mean time at the place of observation 1h 11m 16
Time at Greenwich by chronometer.. 4 3 8

Difference in the longitude........

2 51 52 42° 58′ W., the longitude being

west, because the time at Greenwich is the greatest.

EXAMPLE IV.

Suppose that, on May 14, 1836, sea account, the mean time was found, by an altitude of the sun, to be 3h 59m 09 P. M., when the time by the chronometer was

2h P. M., the chronometer being too slow for mean Greenwich time 11" 9". Required the longitude.

Suppose that, on June 14, 1836, sea account, in a place whose longitude from Greenwich was known, a number of observations were taken to ascertain the going of the chronometer; and it was found, that, on that day, it was 10 too slow for mean Greenwich time, and lost time 2 per day; and that, on July 14, 1836, sea account, the time per chronometer was 6h 0m 6 P. M., when, by an observed altitude of the sun, the mean time was 1 21m 32 P. M. Required the longitude.

Suppose that, on June 15, 1836, in the afternoon, astronomical account, at Boston, in the latitude of 42° 21′ 15′′ N., and longitude 71° 04' 09" W., several angular distances of the sun's lower limb, from its reflected image in a basin of quicksilver, were observed, and the times noted by a chronometer, which was supposed to be very nearly regulated for mean time at Greenwich; the times and altitudes being as below; the thermometer standing at 76°, and the barometer at 30°.05. Required the error of the chronometer relative to mean time at Greenwich.

Times by the chronometer, June 15d 7h 55m 20°....... Observed angle 91°16′ 20′′

Half the mean angle is equal to altitude's lower limb.............
Refraction, Table XII.-57"-Parallax, Table XIV.+6"=—51"
Table XXXVI., Thermometer 76°, correction

-3"

Barometer 30°.05, correction +1"}

Correction for refraction and parallax..

*

= -2

O's semidiameter by Nautical Almanac

's true altitude...

In finding the sun's declication, semidiameter, &c., from the Nautical Almanac, the time at Greenwich is supposed to be the same as the mean time of the observation by the chronometer, 757 238.2, which is supposed to be very nearly regulated to mean time at Greenwich. If you have no chronometer regulated for that meridian, you must estimate the time at Greenwich in the usual manner, by adding to the mean time at the ship, the longitude if west, or subtracting it if east.