4th. The reduced distance, found as above, has been corrected to the spheroidal figure of the earth, according to the theory explained in the Philosophical Transactions of the Royal Society of 1797; and for doing which, rules are given by Mr. Mendoza with his Nautical Tables of 1801. This calculation is tedious, and the correction, more especially in low latitudes, too small to be necessary in common cases.
5th. In the nautical almanack the distances are given to every three hours, but the irregularities of the moon's motion being such as to cause some inequality in the different parts of this interval, the distance at the hour preceding, and at the hour following the time of observation, was found by interpolation from the two nearest given on each side; and having the distances at Greenwich for each hour, the observed distance can never fall more than half an hour from one of them; and the moon's inequalities do not then produce any sensible error in the corresponding time, as obtained from common proportion. The correction arising from this process is seldom so important as to be necessary in sea observations.
6th. The longitude deduced from a comparison of the true distance at observation with the hourly distances at Greenwich, is contained in the following tables under the head of Longitude from Nautical Almanack. But as it frequently happened, that the observation was not taken exactly in the place which it is intended to fix, this longitude is reduced to that place by the application of the difference shown by the time keepers to have existed between the two situations. In ascertaining this difference, the rates of going allowed to the time keepers are generally those found at the place which is to be fixed; whether applied to observations taken before arriving, or after quitting that place. This, however, could be done only at those stations where rates had been observed; at the intermediate points, where the result of lunar distances is given principally as an object of comparison with the time keepers, the rates allowed in the reduction are those found at the station previously quitted; but then the difference of longitude is corrected by the quantity consequent on the following supposition: that the time keepers altered their rates from those at the previous, to those at the following station, in a ratio augmenting in arithmetic progression. The difference of longitude, thus corrected when necessary, is given under the head of Reduction by time keepers; and the longitudes reduced by it to the place intended to be fixed, are taken to be of equal authority with those resulting from observations made in the place itself.
7th. But these longitudes, whether reduced to, or observed in, the place to be fixed, still require a correction which is of more importance than any of those before mentioned. The theories of the solar and lunar motions not having reached such a degree of perfection as to accord perfectly with actual observation at Greenwich, the distances calculated from those theories and given in the almanack become subject to some error; and consequently so do the longitudes deduced from them. The quantities of error in the computed places of the sun and moon, have been ascertained at Greenwich as often as those luminaries could be observed; and Mr. Pond, the astronomer royal, having permitted access for this purpose to the table of errors kept in the Observatory, Mr. Crosley has calculated the corresponding effects on the longitude, and proportioned them to the time when our observations were taken. The combined effect of the two errors forms a correction to the longitudes obtained from the sun and moon; but when the moon was observed with a star, then the moon's error alone gives the correction. But it has sometimes happened, that there were many days interval between the observations of the moon at Greenwich, and that the errors preceding and following are so extremely irregular, that no accuracy could be expected in reducing them by proportion; in these unfortunate cases, that part of the error belonging to the moon has been taken absolute, such as it was found on the day nearest to the time of observation; but the sun's error is always from proportion. These corrections, with the interval in the Greenwich observations of the moon, are given under their proper heads.
8th. The longitudes thus computed, reduced to the intended point, and corrected, are placed under each other; and the mean of the whole is taken to be the true longitude of that point, unless in certain cases where it is otherwise expressed. The mean is also given of the longitudes uncorrected for the errors of the sun and moon's places, that the reader may have an opportunity of comparing them; and some sea officers who boast of their having never been out more than 5', or at most 10', may deduce from the column of corrections in the different tables, that their lunar observations could not be entitled to so much confidence as they wish to suppose; since, allowing every degree of perfection to themselves and their instruments, they would probably be 12', and might be more than 30' wrong.
In the nautical almanacks for 1811 and 1815, the distances are computed from the new tables of Burg for the moon, and of Delambre for the sun; and it is to be hoped that the necessity of correcting for errors in the distances at Greenwich will have ceased, or be at least greatly diminished. Should the computed places of the sun and moon be happily found to agree with actual observation, and supposing that our results may be taken as the average of what practised observers with good instruments will usually obtain when circumstances are favourable, then lunar observations taken in 1814 and afterwards, may be entitled to confidence within the following limits:
From one set of distances, consisting of six independent sights, the error in longitude may be 30' on either side; but will probably not exceed 12'.
From six sets on one side of the moon, each set consisting as above, the error may be 20'; but not probably more than 8'.
Twelve sets of distances, of which six on each side of the moon, are not likely to err more than 10' from the truth; and may be expected to come within 5'.
The error in sixty sets, taken during three or four lunations, and one half on each side of the moon, will not, I think, be wrong more than 5'; and will most probably give the longitude exact to 1' or 2', This degree of accuracy is far beyond what the hopes of the first proposers of the lunar method ever extended, and even beyond what astronomers accustomed only to fixed observatories will be disposed to credit at this time; but in thinking it probable that sixty sets of lunar distances will come within 1' or 2' of the truth, when compared with correct tables, I conceive myself borne out by the following facts.