The lunar tables of Tobias Mayer
Astronomical tables giving the Moon's position at any moment of time to within about a minute of arc: this is the threshold at which the Moon can serve as a clock for finding longitude at sea. An error of one minute of arc is roughly half a degree of longitude, the limit set by the reward of 1714.
Gravitational theory alone was not enough for that. The German astronomer Tobias Mayer took the equations to which Euler had reduced lunar theory and fitted their coefficients to long runs of observations, joining a derivation from theory to a treatment of data. Using his tables the Astronomer Royal James Bradley at Greenwich obtained the Moon's place accurately enough to determine longitude at sea to within half a degree.
The Board of Longitude accepted the tables but judged them unusable at sea as they stood: working through them cost the mariner hours. Mayer's widow received 3,000 pounds from the Board, and the tables themselves became the numerical foundation of the first ten issues of the Nautical Almanac.
Why this date
a spread. The tables were communicated to the Royal Society of Sciences at Gottingen in 1752 and printed in its transactions; a corrected manuscript was sent to the British government in 1755 and compared by Bradley with the Greenwich observations (this version, according to Wepster, was never published); the version of 1762 underlay the first issues of the Nautical Almanac; an improved set appeared posthumously in London in 1770, and the theory behind it, Theoria lunae juxta systema Newtonianum, in 1767. The axis value is 1755, the year of the first documented application, that is of the submission to Britain and Bradley's check: the year of the Gottingen volume could not be verified.
What it grew from
-
Mathematical Principles of Natural Philosophy
influenced · Confidence: medium
Mayer himself names the foundation of his tables: the famous theory of the great Newton, reduced by Euler to general analytic equations. Gravitation supplied the form of the equations, observation supplied the coefficients.
Dispute: The confidence is medium for two reasons. First, the claim about Newton comes from Mayer himself rather than from a historian: Higgitt, speaking of his tables, names Euler and the treatment of observations and does not name Newton at that point. Second, the Royal Museums Greenwich state plainly that the theory of the Principia was not enough for longitude, and that what made the tables work was empirical fitting rather than derivation from theory. The connection exists, but it supplies the form of the equations, not the result.
Link sources
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026, quotation assembled from fragments
I have constructed theses tables ... with respect to the inequalities of motions, from that famous theory of the great Newton ... which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.
- Rebekah Higgitt, «Longitude», Encyclopedia of the History of Science, vol. 4, issue 1 (April 2021) (2021) Verified 20 Sep 2026
One went to the widow of Tobias Mayer, whose application of Leonhard Euler's mathematical techniques to lunar observations had allowed him to produce greatly improved lunar tables.
- «Longitude found: Nevil Maskelyne and the lunar method», Royal Museums Greenwich (2024) Verified 20 Sep 2026
Isaac Newton's theories within his book Principia failed to describe the Moon's motions well enough to find longitude accurately.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026, quotation assembled from fragments
What grew out of it
-
The Nautical Almanac and the lunar-distance method in the mariner's hands
prerequisite for · Confidence: high
The almanac is numbers recast into a form the mariner can use, and those numbers were Mayer's: every lunar position in the first ten issues, for the years 1767 to 1776, was computed from Mayer's tables of 1762.
Dispute: The fact itself is not in dispute, but the division of labour deserves to be stated exactly: Mayer supplied the numbers, while what made the method usable at sea was not the quality of the tables but the removal of the computing to the shore. Wepster states plainly that the Board found Mayer's method itself impractical because of the amount of calculation, so the almanac is not a continuation of the tables but a different move made on their basis.
Link sources
- Steven Wepster, «Precomputed lunar distances», (Universiteit Utrecht) Verified 20 Sep 2026
In the first ten almanacs, for the years 1767-1776, all lunar positions were computed from Mayer's 1762 tables.
- Steven Wepster, «Precomputed lunar distances», (Universiteit Utrecht) Verified 20 Sep 2026
And Mayer's method was not practical because it entailed too much work, in the form of many hours of calculations.
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026
The Tables of the Moon had been brought by the late Professor Mayer of Göttingen to a sufficient exactness to determine the Longitude at Sea to within a Degree, as appeared by the Trials of several Persons who made use of them.
- Rebekah Higgitt, «Longitude», Encyclopedia of the History of Science, vol. 4, issue 1 (April 2021) (2021) Verified 20 Sep 2026
One went to the widow of Tobias Mayer, whose application of Leonhard Euler's mathematical techniques to lunar observations had allowed him to produce greatly improved lunar tables.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Steven Wepster, «Precomputed lunar distances», (Universiteit Utrecht) Verified 20 Sep 2026
Node sources
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026, quotation assembled from fragments
I have constructed theses tables ... with respect to the inequalities of motions, from that famous theory of the great Newton ... which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026
Mayer began calculating lunar and solar tables in 1753 and in 1755 he sent them to the British government.
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026
These tables were good enough to determine longitude at sea with an accuracy of half a degree.
- «Tobias Mayer», MacTutor History of Mathematics Archive, University of St Andrews Verified 20 Sep 2026
The Board of Longitude sent Mayer's widow £3000 as an award for the tables.
- Rebekah Higgitt, «Longitude», Encyclopedia of the History of Science, vol. 4, issue 1 (April 2021) (2021) Verified 20 Sep 2026
One went to the widow of Tobias Mayer, whose application of Leonhard Euler's mathematical techniques to lunar observations had allowed him to produce greatly improved lunar tables.
- Steven Wepster, «Precomputed lunar distances», (Universiteit Utrecht) Verified 20 Sep 2026
And Mayer's method was not practical because it entailed too much work, in the form of many hours of calculations.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.