Sir,

Having long deferred to make any report relating to the observations that were taken at sea by captain Campbell, in the year 1757, which you transmitted to me by order of the lords of the admiralty, I think it necessary to acquaint you, that, upon examining those observations, I perceived that they were not in all respects accompanied with such circumstances as are requisite for forming a right judgment of the accuracy and certainty with which observations proper for finding the longitude at sea by the moon can in fact be taken; for which reason I delayed giving my opinion upon this point till I could have an opportunity of comparing a greater variety of observations, made at different times, and with different instruments: such an opportunity having lately been given me by captain Campbell, who has favoured me with a copy of several observations that were made by him in 1758 and 1759, I now beg leave to lay before their lordships the result of the comparisons which I have made.

But before I proceed farther it may be proper to take notice, that since the time when I gave their lordships an account of the near agreement of Mr. Professor Mayer's lunar tables with the observations that had been then made at the Royal Observatory, I have compared several others, which concurred to prove that the difference between the observed and computed places nowhere amounted to more than about one minute and a half; and I find that the difference (small as it is) may yet be diminished by making alterations in some of the equations, whose true quantity could not be determined without proper observations; after making the needful corrections it appeared, by the comparison of above eleven hundred observations taken here since the new instruments were fixed up, that the difference did nowhere amount to more than a minute: it may therefore be reasonably concluded, that so far as it will depend upon the lunar tables the true longitude of a ship at sea may in all cases be found within about half a degree, and generally much nearer.

This fundamental and most important article being established upon such full evidence, it remained to be examined within what limits the errors arising from observations actually taken at sea could be contained.

In order to determine this point, I computed the ship's longitude from each of the observations made by captain Campbell, and, upon comparing the results of several that were taken near the same time and under the like circumstances, it appeared that in general the observer was not liable to err more than one minute in judging of the apparent contact of the moon's limb and the object with which it was compared. Now this being nearly the same error that would be found to obtain, if the like observations were to be made with the same instruments on land, it may hence be inferred, that in moderate weather the motion of the ship is no otherwise an impediment in this sort of observations, than as it renders the repetition of them more tedious and troublesome to the observer, which yet ought by no means to be omitted; because if each single observation be liable to an error of a minute only, by taking the mean of five or six, the error on this head may be so far diminished as to be of small moment.

I would by no means attempt to infer from hence, that the longitude found by observations of this sort may in all cases be depended upon within one degree; but I beg leave to observe, that whatever extraordinary circumstances may have concurred to produce so near an agreement in this particular case, the event is such as may give reason to hope, however great the difficulties of finding the longitude by this method seem to be, that they are not insuperable, or such as ought to deter those whom it most nearly concerns from attempting to remove the