It is evident, however, that chronometers are liable to a variety of accidents, and that in very long voyages the means of verifying their rate of going seldom occur.  Hence the lunar method, or the method of ascertaining the longitude by means of the motions of the moon, is more useful and valuable.  Here again, the profoundest researches of Clairaut, Euler, D’Alembert, and La Place, were brought practically to bear on navigation.  Guided and aided by these, Tobias Mayer, of Gottingen, compiled a set of solar and lunar tables, which were sent to the lords of the admiralty, in the year 1755; they gave the longitude of the moon within thirty seconds.  They were afterwards improved by Dr. Maskelyne and Mr. Mason, and still more lately by Burg and Burckhardt; the error of these last tables will seldom exceed fifteen seconds, or seven miles and a half.  The computations, however, necessary in making use of these tables, were found to be very laborious and inconvenient; to obviate this difficulty, the nautical almanack, suggested by Dr. Maskelyne, was published, which is now annually continued.  The longitude is thus ascertained to such a nicety, as to secure the navigator from any danger arising from the former imperfect modes of finding it; “he is now enabled to make for his port without sailing into the parallel of latitude, and then, in the seaman’s phrase, running down the port, on the parallel, as was done before this method was practised.  Fifty years ago, navigators did not attempt to find their longitude at sea, unless by their reckoning, which was hardly ever to be depended on.”

Not long after the mariner’s compass was employed, its variation was noticed; as it is obvious that, unless the degree and direction of this variation are accurately known, the compass would be of little service in navigation, the attention of navigators and philosophers was carefully directed to this point; and it was ascertained that the quantity of this variation is subject to regular periodical changes.  By means, therefore, of a table indicating those changes, under different latitudes and longitudes, and of what are called variation charts, the uncertainty arising from them is in a great measure done away.  Another source of error however existed, which does not seem to have been noticed till the period of Captain Cook’s voyages:  it was then found, “that the variation of the needle differed very sensibly on the same spot, with the different directions of the ship’s head.”  Captain Flinders attributed this to the iron in the ship, and made a number of observations on the subject; these have been subsequently added to and corrected, so that at present the quantity of variation from this cause can be ascertained, and of course a proper allowance made for it.  It does not appear that any material improvement has been made in the construction and use of the log,—­that useful and necessary appendage to the compass,—­since it was invented about the end of the sixteenth century.