Essay Review: Tobias Mayer's Lunar Theory: Between Theory and Observations: Tobias Mayer's Explorations of Lunar Motion, 1751–1755

(ProQuest: ... denotes formulae omitted.)TOBIAS MAYER'S LUNAR THEORY Between Theory and Observations: Tobias Mayer's Explorations of Lunar Motion, 1751-1755. Steven A. Wepster (Springer Verlag, Berlin, 2010). Pp. xiv + 246. euro96. ISBN 978-1-4419-1313-5.Tobias Mayer (1723-62), professor of practical mathematics at the Georg-August Academy in Gottingen from 1751 till his death in 1762, published in 1753 tables of the Moon's motion which, in comparisons with 200 observations, differed in all but a dozen cases by less than \\ arcminutes. Consequently, deteimining longitude at sea by the method of lunar distances looked to be within reach: an accuracy of 2 arcminutes would enable the longitude to be computed to Io. Leonhard Euler among others encouraged the modest young Mayer to compete for the Longitude Prize set up by the British Parliament, promising rewards of £20,000, £15,000, and £10,000 for procedures giving the longitude to \°, §°, and Io, respectively. Mayer's final tables, submitted by his widow in 1763, became the initial basis for the British Nautical almanac and astronomical ephemeris, which began publication in 1767 and has appeared annually ever since.How did Mayer obtain lunar tables good to within 2 arcminutes, when the theories derived from Newton's gravitational law by the leading mathematicians of the day - Euler, Clairaut, and d'Alembert - erred frequently by 3 to 5 arcminutes? Mayer did not conceal that he had honed the accuracy of his tables by comparisons with observations, but of his process he revealed no details. Conjectures about it put forward later have remained unsupported by solid evidence. Gauss, believing the method of least squares the obvious way to go, suggested that Mayer had used it (he would thus have been the first to do so), but on looking into Mayer's papers found only "home-baked combinations [hausbackene Combinationen]". Evidently, any understanding of Mayer's success must be extracted from Mayer's worksheets, extant in some twenty-four manuscript codices now preserved in Gottingen. But can these manuscripts be made to divulge their story? This is Steven A. Wepster's project in the study here reviewed.An initial step was to sort out the different tasks accomplished in the worksheets, and to order them chronologically. Numerous among the worksheets are position computations: lunar positions computed from lunar tables for specified times and compared with observations. Codex //*,, for example, contains 366 such computations. The ostensible aim is to check the accuracy of lunar tables against observations, assuming the accuracy of the latter. Since during the years 175 1-55, the period of Mayer's most intense work on the tables, the new Gottingen observatory was incomplete, lacking its mural quadrant, Mayer had to depend chiefly on observations recorded by other astronomers. He sought observations of high quality, rejecting lunar longitudes derived from meridian transits, which were frequently in error because of erroneous stellar longitudes assumed in their derivation. In the determinations most crucial for the tabular accuracy he achieved, Mayer employed occultations of stars by the Moon. The winking out of a star as it is eclipsed by the Moon and its reappearance when uncovered are among the most sharply defined of celestial events. At the moment of immersion or emersion, a point on the Moon's limb has the same longitude as the star, and this fact could be used to investigate the Moon's inequalities without adopting any pre-assigned value for the star's longitude.To identify the theory under test in Mayer's position computation sheets, Wepster had to depend on numbers that the sheet contained: 'mean motions' in the values of longitude, apogee, or ascending node; 'epochs', the values of these parameters for fixed dates; and periodic departures from the mean motions. Traditionally, astronomers called the periodic departures "inequalities" and modelled them by circular functions called "equations". …