4. But where a transit instrument cannot be carried, or cannot be used, an at sea, the longitude must be found astronomically by the distance of the moon from the sun, planets, or fixed stars, measured with a reflecting instrument. This apparent distance is reduced to the true distance, that is, such as It would be, seen from the centre of the earth, and as these distances are computed and set down iu the ' Nautical Almanac' for every three hours Greenwich time, as they would be seen from the same place, the Greenwich time corresponding to the time of the observation can be calculated. But the time at the place is always supposed to be known from observation, and hence the difference gives the longitude. The longitude may be determined on shore by lunar observations, and, if a gelid be used, with much greater accuracy than at sea. All ships and travellers ought to be well supplied with chronometers, that is, the means of keeping their Greenwich time when by observation they have got it, and then the result of the observe Goa and computation is simply stated to be the error of the chronometer on Greenwich time. The chronometer, if the into be pretty well Irene n, continues to give the Greenwich time (the correction for error and rate being opplieel) for several days; and the longitude is found every day, by oompoxing the actual time at the place of observation with the Greenwich time at the ammo moment, given by the chrono meter. NVer have spoken as if one chronometer alone were used, but it is mere folly to rely upon one or even two chronometers in a ship, or in important geographical rewoirches.• Those are to he compared from day to day, to ascertain that they ars not emdlenly altering their rates, and also whenever any warunentioal observation is made which deter mines the Greenwich time (for that gives the error of each of the or the time at the phiee. In reducing observed lunar distances to the true lunar distances, the altitudes of the sun and moon. or moon and stars at the time of observation are required, and at sin two oleerven, are commonly set to measurer these altitudes at the moment the lunar observer gives a signal that lie has made the rontact; indeed a fourth person is sometimes engaged in noting the chronometer. On shore this profusion of aide cannot always be ob tained, nor are they at all wanted. If the time at the place and the latitude be known, the altitudes may be computed, or the observer may proceed thus : 1st, an altitude of the sun, planet, or star ; an altitude of the moon's bright limb; 2rdly, three lunar distances; lthiy, a second altitude of the moon ; and ;Ably, a second altitude of the star or sun, noting the chronometer at each observation. Ile will then have the means, by simple proportion, of reducing the altitudes to what they would have been at the time of observing the lunar distances.
We should advise observers, who are properly furnished with chrono meters, rather to make a large ntunber of observations on a few favour able nights, than to take a few observations on many nights. By observing several lunar distances on both sides of the moon, and from all the stars and planets east and west given in the 'Nautical Almanac,' the errors of the instrument may be in a great measure eliminated, and the error of observation much diminished. It Is evident that if two equal distances are taken, one east of the moon and the other west, then any error of the Instrument, such as erroneous index error, want. +A parallelism in the glasses or telescope, excentrieity, &c., would be the same in each, and therefore could be got rid of. In like manner, if two observations on the same side of the moon give different longi tudes, it is clear that the instrument has somc'error which is not index error. Ou this subject we shall have occasion to speak again in treating of the sextant. The luni-solar observations are generally preferred by seamen (and they are perhaps the most satisfactory), partly perhaps because the altitude of the sun, by giving time at the place, is ianne diately applicable to the determination of the longitude. The longi tude from lunar distances, however carefully taken, cannot be relied upon to very great nicety. With all appliances, a distance to 10% and a longitude to 20', or Ath of a degree, can scarcely be considered as certain, and the errors of the lunar tables will not unfrequently double this error. At sea it would not be safe to rely on any series of lunar distances for a less quantity than 1m of longitude, or 1th of a degree, but this is quite sufficient for the purposes of navigation in open seas.
The computation of lunar distances is very easily performed by Thomson's tables, which are exceedingly convenient, and require only a little more extension. They are approximate, but near enough for the navigator and the traveller en route. As the last accuracy can only be given to the computations after the errors of the lunar table* are corrected, there is no need of much refinement in the previous work.
It has been proposed to determine the longitude on shore by taking altitudes of the moon with the mercurial horizon ; and between and near the tropics the method may be a good one. In these low latitudes the motion of the moon in altitude is nearly vertical and very rapid, and this motion is doubled by observing the distance between the moon and its image seen by reflection. The bright edge of the moon is a good object in reflecting instruments. The calculation may be made thus : The time at the place must be most scrupulously determined, and the error and rate of the chronometer known, if possible, by equal altitudes of the sun on each side of noon and midnight. Then, the error of the chronometer being known at the place and time of observation, assume an approximate value of the longitude, and deter mine the Greenwich time, the It. A. and declivatiou of the moon, and its apparent diameter and parallax. Each altitude of the moou will, when properly reduced, give an error of the chronometer, which, if tho assumed longitude be correct, will agree with the error obtained by the sun or stars. If there be a difference, a second longitude may be assumed, and thus, as has been shown in speaking of transits of the moon, the terms may be obtained from which, by a simple proportion, the true longitude can be deduced. It is not improbable that the doubling the motion of the moon in observing by reflection, and the sharpness of the images, may make up for the error committed in ascertaining and keeping the time; but of this the observer must be the judge.
Ikternanation of Oreenteielt Time by the Greenwich, time has been extracted from astronomical phenomena, but where the distance is not great, the time may be brought from Green wich by chronometers. Suppose, for instance, the longitude of Madeira were required : then, having ascertained the errors and rates of several good chronometers at Greenwich, they are carried to Madcira,aud their errors on the meridian of Madeira, and their rates, determined there. The Greenwich time is known from each chronometer, supposing tho rate during the voyage to be the mean of the rates before and after, and thus each chronometer gives a longitude of Madeira, and the mean of the whole Is taken. The voyage back to Greenwich, in like manner, with the errors and rates on arriving at Greenwich, furnish a second longitude; and if the motion at see has any toudency to alter the rates, this cause will affect the first determination with a certain error, while it affects, the second determination with exactly the same error, but in a different sense. If it increases the difference of longitude one way, it rlieninishea it the other way. By a mean of several such voyages tho diffenmee of longitude of places within a week or perhaps a fortnight's journey may be ascertained with considerable exactness. There arc some precautions to be adopted in determining longitudes chrono metrically, which ought not to be neglected. It is well known that two observers will sometimes differ several tenths of a second from each other in getting the time at the same place and with the same instrument. Now this personal equation, if it exist between the observers at the two places, will affect the longitude by exactly its amount : hence the observers should be reversed for half the time of the experiment, if possible, or their relative personal equation found by comparison with each other, or at least with a third person. • The accuracy of a chronometrical longitude depends on the distance in time between the places compared and the smoothness of the conveyance. It is decidedly the best mode where the distances do not exceed three or 'four days' journey, and where there are good carriage roads or 'steam-boats. At sea, for voyages of moderate length, the Greenwich time may be taken almost entirely from chronometers ; and if the number be considerable, and the watches good, there is little occasion for lunar distances, except for the greater caution, and to be assured against any accident affecting all the timekeepers the same way.