Abstract
Three years after Dunthorne published the first detailed study of the moon’s secular acceleration, Tobias Mayer produced the first set of lunar tables which incorporated the secular acceleration directly into the calculation of the moon’s longitude. Dunthorne had provided a table that allowed a correction to be applied to lunar positions calculated from his lunar tables. Dunthorne’s correction, however, was applied after the moon’s position had been calculated. By contrast, Mayer integrated the secular acceleration within the initial calculation of the moon’s mean position. This meant that the effect of the secular acceleration was taken into account when determining the various equations of anomaly (which depend upon the elongation of the mean moon from the sun and the mean moon from the apogee). The difference between applying the correction for the secular acceleration before or after calculating the anomaly is small but not trivial, and Mayer’s method was the theoretically correct one. Mayer adopted a value for the size of the secular acceleration that was considerably smaller than that found by Dunthorne. Mayer did not explain how he had derived this value in the introduction to his published tables, but it is possible to reconstruct his general method from his preserved manuscript notes.
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Notes
- 1.
The following is based upon the detailed biography by Forbes (1980). See also Forbes (1967) and Wepster (2010), pp. 27–42.
- 2.
Forbes (1980), p. 36.
- 3.
Forbes (1980), pp. 42–43.
- 4.
Forbes (1980), pp. 106–133.
- 5.
Mayer to Delisle 14 January 1751; see Forbes (1983), no. 9.
- 6.
Forbes (1971).
- 7.
Euler to Mayer 11 June 1754; translation by Forbes (1971), p. 86.
- 8.
Euler to Mayer 11 June 1754; translation by Forbes (1971), p. 86.
- 9.
Wepster (2010) denotes these tables by the code “kil”.
- 10.
A list of Mayer’s lunar tables (including cases of known table parameters and equations which may not have eventually been used by Mayer to draw up tables) is given by Wepster (2010), p. 212.
- 11.
Wepster (2010), pp. 210–211.
- 12.
Wepster (2010).
- 13.
Forbes (1971), p. 65.
- 14.
Euler explains the origin of the observations given to Mayer by Schumacher in a letter dated 15 May 1753. See Forbes (1971), p. 68.
- 15.
The following discussion is based upon the work of Wepster (2010).
- 16.
Mayer to Euler 7 May 1753. See Forbes (1971), pp. 65–66.
- 17.
Euler to Mayer 26 February 1754; translation by Forbes (1971), p. 79.
- 18.
Bevis, “Mayer’s new Tables of the Sun and Moon”.
- 19.
Forbes (1980), p. 142.
- 20.
Bevis, “Mayer’s new Tables of the Sun and Moon”, p. 376.
- 21.
The Gentleman’s Magazine 24 (September 1754), p. 439.
- 22.
The table as printed contains one typographical error: the correction for A.D. 100 is given as 0°38′35″ instead of 0°28′35″.
- 23.
Mayer, “Novae Tabulae Motuum Solis et Lunae”, pp. 388–391. In making this translation I have often referred to Bevis’s English paraphrase of the passage in his letter to The Gentleman’s Magazine of 1754.
- 24.
Wepster (2010), p. 151.
- 25.
Halley’s reconstruction is given in his “Emendationes ac Notæ in vetustas Albatênii Observationes Astronomicas, cum restitione Tabularum Lunisolarium eiusdem Authoris”, p. 920.
- 26.
It is possible that the 6″″ per year comes from comparing the mean motion over 60 years for al-Battānī’s epoch and Mayer’s revised value of the mean motion of 1S10°43′30″ for A.D. 1700. This would give an acceleration of 5;55″″ per year which would naturally round to 6″″ per year.
- 27.
MS Mayer 1541. ff. 140v–143v, 114r–146r, 155v–156v, 159r–166r.
- 28.
MS Mayer 1541, f. 160r.
- 29.
MS Mayer 1541, f. 140v.
- 30.
MS Mayer 1541, f. 159r.
- 31.
MS Mayer 1541, f. 164r.
- 32.
Wepster (2010), pp. 143–176.
- 33.
Mayer to Euler 25 November 1753; translation by Forbes (1971), p. 77.
- 34.
MS RGO 4.125, p. 29 and an unpaginated sheet.
- 35.
Euler, “Concerning the Contraction of the Orbits of the Planets”, pp. 356–357.
- 36.
Mayer, “Novae Tabulae Motuum Solis et Lunae”, pp. 391–392.
- 37.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 73.
- 38.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 74.
- 39.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 74.
- 40.
Mayer to Euler 22 August 1753; translation by Forbes (1971), pp. 74–75.
- 41.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 75.
- 42.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 75.
- 43.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 75.
- 44.
Mayer to Euler 22 August 1753; translation by Forbes (1971), p. 76.
- 45.
Euler to Mayer 26 February 1754; translation by Forbes (1971), p. 79.
- 46.
MS Mayer 9; edited by Forbes (1972).
- 47.
Mayer’s copies of these letters are preserved in MS Mayer 152, ff. 1r–3v.
- 48.
MS Mayer 152, f. 1v.
- 49.
MS Mayer 152, f. 2r.
- 50.
Euler to Mayer 26 February 1754; translation by Forbes (1971), pp. 79–80.
- 51.
Mayer to Euler 6 March 1754; translation by Forbes (1971), pp. 84–85.
- 52.
MS Mayer 9; edited by Forbes (1972).
- 53.
MS Mayer 9, § 54; edited by Forbes (1972), p. 97.
- 54.
MS Mayer 9, § 55; edited by Forbes (1972), p. 97.
- 55.
Robert Newton has called this approach “playing the identification game” and demonstrated how it results in unjustifiable confirmation of existing parameters. See Newton (1969).
- 56.
Forbes (1972), pp. 17–18.
- 57.
Forbes (1972), p. 17.
- 58.
MS Mayer 9, § 51; edited by Forbes (1972), p. 95.
- 59.
MS Mayer 9, § 51; edited by Forbes (1972), p. 95.
References
MS Mayer 1528
MS Mayer 1533
MS Mayer 1541
MS Mayer 1548
B. J. [John Bevis], 1754, ‘Mayer’s new Tables of the Sun and Moon’, The Gentleman’s Magazine 24, 374–376.
Euler, Leonard, 1746, Opuscula varii argumenti (Volume 1, Berlin).
Mayer, Tobias, 1753, ‘Novae Tabulae Motuum Solis et Lunae’, Commentarii Societatis Regiae Scientiarum Gottingensis 2, 383–430.
Forbes, E. G., 1967, ‘The Life and Work of Tobias Mayer (1723–62)’, Quarterly Journal of the Royal Astronomical Society 8, 227–251.
Forbes, E. G., 1971, The Euler-Mayer Correspondence (1751–1755) (New York: Elsevier).
Forbes, E. G., 1972, The Unpublished Writings of Tobias Mayer. Vol. I: Astronomy and Geography (Göttingen: Vandenhoeck & Ruprecht).
Forbes, E. G., 1980, Tobias Mayer (1723–62): Pioneer of Enlightened Science in Germany (Göttingen: Vandenhoeck & Ruprecht).
Forbes, E. G., 1983, ‘La Correspondance Astronomique entre Joseph-Nicolas Delisle et Tobias Mayer’, Revue d’Histoire des Sciences 36, 113–151.
Newton, R. R., 1969, ‘Secular Accelerations of the Earth and Moon’, Science 166, 825–831.
Wepster, S., 2010, Between Theory and Observations: Tobias Mayer’s Explorations of Lunar Motion, 1751–1755 (New York: Springer).
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Steele, J.M. (2012). An Integrated Approach: Tobias Mayer. In: Ancient Astronomical Observations and the Study of the Moon’s Motion (1691-1757). Sources and Studies in the History of Mathematics and Physical Sciences. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-2149-8_7
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