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§§ 225, 226] Much more important, however, were his lunar theory and the tables based on it. The intrinsic mathematical interest of the problem of the motion of the moon, and its practical importance for the determination of longitude, had caused a great deal of attention to be given to the subject by the astronomers of the 18th century. A further stimulus was also furnished by the prizes offered by the British Government in 1713 for a method of finding the longitude at sea, viz. £20,000 for a method reliable to within half a degree, and smaller amounts for methods of less accuracy.

All the great mathematicians of the period made attempts at deducing the moon's motions from gravitational principles. Mayer worked out a theory in accordance with methods used by Euler (chapter, § 233), but made a much more liberal and also more skilful use of observations to determine various numerical quantities, which pure theory gave either not at all or with considerable uncertainty. He accordingly succeeded in calculating tables of the moon (published with those of the sun in 1753) which were a notable improvement on those of any earlier writer. After making further improvements, he sent them in 1755 to England. Bradley, to whom the Admiralty submitted them for criticism, reported favourably of their accuracy; and a few years later, after making some alterations in the tables on the basis of his own observations, he recommended to the Admiralty a longitude method based on their use which he estimated to be in general capable of giving the longitude within about half a degree.

Before anything definite was done, Mayer died at the early age of 39, leaving behind him a new set of tables, which were also sent to England. Ultimately £3,000 was paid to his widow in 1765; and both his Theory of the Moon and his improved Solar and Lunar Tables were published in 1770 at the expense of the Board of Longitude. A later edition, improved by Bradley's former assistant Charles Mason (1730–1787), appeared in 1787.

A prize was also given to Euler for his theoretical work; while £3,000 and subsequently £10,000 more were awarded to John Harrison for improvements in the chronometer,