Seldom Amounting to Twenty Seconds
In 1766 the Astronomer Royal promised that the Moon tables behind his new Nautical Almanac were seldom wrong by 20 seconds of arc, so a navigator's longitude would generally be good to 10 miles. Here are all 1,816 lunar distances printed for four months of 1767, transcribed twice and checked against where the Moon really was.
That paragraph is four promises, and each is a number. The almanac was new: the first British one, published in 1766 for the year 1767, and built so that a ship's officer with a sextant could find longitude from the Moon. The Moon crosses the sky against the stars at about its own width every hour, so its distance from a bright star is the hand of a clock that reads the same for everyone on Earth. The almanac printed where that hand would point, every three hours, by the clock at Greenwich. Measure the distance at sea, look it up, and you know Greenwich time; the difference from your local time is your longitude, fifteen degrees to the hour.
Which means every error in a printed distance is an error in time, and every error in time is an error in position. This page takes the promise at its word.
Work a lunar
Pick a real entry from the 1767 table. You are at sea at the moment it was computed for, and your sextant, once cleared of refraction and parallax, reads the true distance plus whatever error your hands add. The page reads Greenwich time off the printed table, by proportional parts as the almanac instructs, and tells you what the printed number cost you.
Held perfect on purpose: your local time, and the clearing of the observed distance. In practice those, and the sextant, were usually the larger errors; this instrument isolates the one the almanac contributed. Miles are nautical miles along your parallel: a minute of longitude at the equator, fewer as you go north or south.
Every distance, checked
Each dot is one printed distance: January, April, July and October 1767, the Moon against the Sun and ten bright stars, every three hours. Its height is the printed value minus the true one, the true one computed from NASA JPL's DE440 ephemeris for the same instant of Greenwich apparent time. Tap a dot to put it in the instrument above.
The promise, clause by clause
| Maskelyne, 1766 | Measured, 1767 |
|---|---|
| “scarce ever exceeding 1′ at the most” | Of the Moon's own printed longitudes, 27 of 246 were more than a minute out, the worst by 78.1″. The distances fared better: 42 of 1,815 over a minute, the worst 67.7″. |
| “and seldom amounting to 20″” | Not in 1767. 65% of the printed Moon longitudes and 50.2% of the printed distances were 20″ or more out. The distances' root-mean-square error is 30.3″. |
| “the Uncertainty … can scarcely exceed half a Degree” | Close to true. From the table's error alone, 70 distances would have put a ship more than 30′ of longitude out; the worst, 36′. (One misprint, below, is the exception.) |
| “and generally will not exceed 10 Miles” | A coin toss at the equator: the median error is 9.8′ of longitude and 48.6% exceed 10′. At 0° latitude, … of them exceed 10 nautical miles (set your latitude in the instrument). |
“Miles” is read as nautical miles, one minute of arc of a great circle, the sea mile of the day. The 1′ and 20″ are about “the Error of Mr. Mayer's last lunar Tables”, that is, the Moon's place, so they are scored first against the Moon longitudes the almanac itself prints, then against the distances the navigator used.
Whose error was it?
A printed distance can be wrong for two reasons. The Moon theory can put the Moon in the wrong place: these were the tables of Tobias Mayer of Göttingen, the best in the world, for which the Board of Longitude paid his widow £3,000. Or the arithmetic can be wrong: every one of these numbers was worked by hand, with logarithms, by hired computers.
The two can be separated, because the almanac also prints the Moon's place itself, its longitude and latitude at every noon and midnight. Compare those with DE440 and you have Mayer's error directly. It has a shape: the printed Moon runs ahead of the real one in every one of the four months, on average by 19.5″ in January, 8.8″ in April, 25.9″ in July and 37.1″ in October, wandering by tens of seconds within each. Its latitude is better, 9.3″ root-mean-square.
Now push that error through to each distance: move the Moon by exactly the amount the almanac's own Moon page is wrong, and ask how far that alone moves the distance. Press Take away Mayer's Moon on the chart above and watch. The scatter collapses from 30.3″ to 6.3″ root-mean-square: 96% of the variance in the printed distances was the Moon theory. What remains is the arithmetic, and the arithmetic was very good, about 6.1″ where the Moon was 30° or more from its star, a little rougher (7.9″) at the closer approaches, where the distance curves fastest.
The errors also give away how the tables were made. If the printed Moon longitudes had been reckoned from the mean equinox, the residual would be 18.3″, not 6.3″: they include the nutation Bradley had published in 1748. If the star places had left out the aberration of light, it would be 13.8″: the computers applied it. Before the Moon is taken away, that second question has no answer at all (29.9″ against 30.3″, the wrong way round), which is a small lesson of its own: a real effect can hide under a bigger error until the bigger error is named. And the Moon's printed error is error against the stars, not a frame the two shared: fitting both at once gives 0.98 of the predicted effect and at most 3.5″ of common offset.
Three misprints
The differences between successive entries are how the almanac's own examiners caught mistakes, and they catch them still. Three places in these pages are printed wrong in a way the page itself gives away.
The transposed Pollux digits are set aside from the statistics above (they are no measure of Mayer or of the arithmetic) and kept on the chart, off its scale. A careful navigator comparing successive differences, as the examiners did, would have seen the jump. One who did not would have been nearly seven degrees wrong.
How this was checked
Transcription, twice. Sixteen pages of distances and four of Moon places were read from the Internet Archive's scan by two separate transcribers who did not see each other's work, then compared, which is the almanac's own method turned on itself. Of 7,416 cells, 1 disagreed: an ink blot over one digit.
Differences. Every run of five consecutive entries is tested by its fourth difference. Every place that fails was re-read on the scan and is recorded with what it shows. As a hostile test, a 5″ slip planted in each cell in turn is caught 91% of the time, so a transcription slip of that size or larger is unlikely to have survived in both copies unseen.
The true values. The Moon and Sun from JPL DE440 with Skyfield; the stars from the Hipparcos new reduction, carried back 224 years by their proper motions (Pollux alone has moved 140″); apparent places with nutation and aberration; the almanac's Greenwich apparent time, on its astronomical day that begins at noon, converted by the equation of time; ΔT about 20 s. A second route, JPL Horizons for the Moon and Sun and ERFA for the stars, was run on 48 random entries and agrees to 0.4″. The errors being measured are tens of seconds.
What this does not establish. Only January, April, July and October were read; the other eight months are unchecked. The navigator in the instrument is idealised. “Miles” is an interpretation. The star named β Capricorni is taken to be the brighter component of that wide pair.
Download the complete check, unzip it into an empty directory, then run:
node research/seldom-amounting-to-twenty-seconds/verify.mjs
Node 22 or later, no packages. It re-derives the reconciled table from the two transcriptions, recomputes every number on this page from the transcriptions and the committed modern positions, reruns the difference test and the hostile fixture, and fails if this page says anything the data do not. The archive also holds the two Python programs that made the modern positions (they need Skyfield, astropy and a 115 MB JPL file) and the scan leaf of every entry. The program alone is also published, to read, at /checks/research/seldom-amounting-to-twenty-seconds/verify.mjs; it needs its data files beside it, which is what the archive is for.