Chapter 3 Chronometers, the equation of time and the Moon
Longitude is a clock problem
Earth turns 15° an hour. Know the time at Greenwich and the time where you are, and the difference is your longitude. The hard part, for most of history, was carrying Greenwich with you.
Latitude is written in the sky: the pole stands as high above your horizon as you are far from the equator, and a single sight of Polaris or the noon Sun tells you how far north you are (Latitude from the sky). Longitude is not written anywhere. Every meridian sees the same sky, only at a different hour. To know how far east or west you are, you have to know what time it is somewhere else.
That one sentence holds the whole problem. Earth turns once a day, so the Sun crosses a meridian 15° west of you one hour after it crosses yours. If you can see local time from the sky and carry the time at a reference meridian with you, the difference between them is your longitude. Seeing local time is not the hard part. Carrying the reference time across an ocean, through heat and cold and a pitching hull, is.
Two clocks and a turning Earth
The figure below looks down on Earth from above the North Pole. Sunlight comes from the right, so the meridian facing the Sun is at noon and the one opposite is at midnight. Drag your meridian around the pole, or press Play to let Earth turn. Notice that the ring of hours stays fixed to the Sun while the continents slide under it. Then look at the two clocks: one keeps Greenwich time, the other shows the Sun’s time where you are.
Here is why it works. Relative to the Sun, Earth turns 360° in one mean solar day of 24 hours:
So if your Sun clock reads 12:50 while Greenwich reads 09:30, you are 3 h 20 min ahead of Greenwich, and
A local time ahead of Greenwich means you are east; behind means west. The rule does not care which meridian you call zero. Greenwich is a convention, recommended for international use by a conference in Washington in October 1884;[1] navigators of other nations long reckoned from Paris, Cadiz or their own capitals, and the arithmetic is the same. Notice also what the clocks are measuring: minutes of time, which are not minutes of arc. One minute of time is 15′ of longitude. One second of time is 15″, about a quarter of a nautical mile at the equator.
Why the problem was so hard
Astronomers had found longitude differences on land for a very long time. The trick is to have two observers time the same sky event, each by local time, and compare notes afterward. On 24 May 997 al-Bīrūnī timed a lunar eclipse at Kāth, in Khwarazm, while Abu’l-Wafā’ timed the same eclipse in Baghdad; the difference between their local times gave the difference in longitude between the two cities.[2] documented In the seventeenth century, observers on land did the same with the eclipses of Jupiter’s moons, which happen often and can be predicted.[3]
None of that worked on a ship. Eclipses of the Moon are rare. Jupiter’s moons need a long telescope held steady, which a rolling deck does not allow. And comparing notes afterward is no help when you need your position tonight. A ship needs the reference time now, on board. Until the late eighteenth century the usual answer was dead reckoning: course from the compass, speed from the log line, time from a sandglass, all added up from the last known point. Errors accumulated with every day out of sight of land.
The loss of Sir Cloudesley Shovell’s fleet on the Isles of Scilly on the night of 22 October 1707, with more than 1,300 men, is often told as the disaster that proved the need for longitude.[4] The evidence is more mixed. When W. E. May worked through the fleet’s surviving logbooks in 1960, he found errors in latitude, compass variation, currents and the charts themselves, not a longitude failure alone.[5] Royal Museums Greenwich, which holds the only contemporary print of the wreck, notes that the cause “remains the subject of debate” and that there is no hard evidence that the disaster prompted the Longitude Act.[4] contested It belongs in the story as a reminder of what navigational error cost, not as the cause of what came next.
What came next was the Longitude Act of 1714. Parliament offered £10,000 for a method that found longitude at sea within one degree, £15,000 within two-thirds of a degree, and £20,000 within half a degree, judged on a voyage to the West Indies, and appointed Commissioners, later called the Board of Longitude, to judge claims and fund promising work.[6][7] documented Half a degree is 2 minutes of time. Across a six-week voyage, that meant a clock that drifted by less than about 3 seconds a day, or an astronomical method good to a minute of arc. Both seemed nearly impossible in 1714. Both were achieved within fifty years.
Break a chronometer
The clock route is the simpler idea. Set a timekeeper to Greenwich before you sail, find local time from the Sun each day, and subtract. The difficulty is that every clock is wrong, and the errors pile up. What matters is not that a clock gains or loses but that it does so predictably. A navigator measures a chronometer’s daily rate in port and applies it at sea. Only the part of the error that the rate does not predict, plus any error in the starting setting, becomes longitude error.
Try it. Set a rate error, a starting error and some day-to-day wander, choose a landfall, and watch what happens on the chart.
Notice three things. First, a steady rate error grows linearly: 1 second a day for 46 days is 46 seconds, or 11.5′ of longitude. Second, the random part grows more slowly, roughly as the square root of the days, which is why a well-behaved chronometer with an honest rate could cross an ocean and still meet the Act’s strictest limit. Third, the conversion from degrees to distance depends on where you are. The east–west distance, called departure, is
Harrison’s timekeepers, and the Board
John Harrison, a carpenter and clockmaker from Lincolnshire, brought his first sea clock to London in 1735. H1 was a large, spring-driven clock with linked balances that cancelled the effect of the ship’s motion; on a trial voyage to Lisbon and back in 1736 it identified the landfall more correctly than the ship’s officers did.[8] documented The Board did not reject him. In 1737 it agreed to pay him £500, half in advance, for an improved version.[8] H2 followed, then H3, begun in 1740, which occupied him for 19 years.[8]
H4 broke with all of them. It is a large watch, not a clock, whose fast-beating balance borrows from a watch Harrison had commissioned from John Jefferys.[8] On its first trial, carried by Harrison’s son William on HMS Deptford from Portsmouth to Jamaica in 1761–62, it was found about 5 seconds slow on arrival once its rate was allowed for: an error of about 1¼′ of longitude.[9][8] documented On a second, more carefully controlled trial to Barbados in 1764, with the astronomer Nevil Maskelyne sent ahead to fix the island’s longitude by observation, H4 erred by 39 seconds after about seven weeks, under 10 miles.[9] Both results lay within the half-degree limit.
Popular accounts often stop there and cast the Board as villains who cheated a lone genius. The documents tell a more complicated story.[10][7] A single watch performing well twice proved that Harrison could make one; it did not prove that his method could equip a navy. The Board’s condition, written into a new Longitude Act of 10 May 1765, was that Harrison explain the watch’s principles and that other makers be able to reproduce it.[8] He received £10,000 once he had demonstrated the principles of H4, and Parliament granted him a further £8,750 in 1773 after a long dispute.[9] documented Harrison was treated stubbornly, and he was stubborn himself. Both statements are true.
The Board’s test of reproducibility was Larcum Kendall’s copy of H4, called K1. James Cook took it on his second Pacific voyage in HMS Resolution, 1772–75, and its performance there persuaded the skeptics; the Arnold timekeepers on the same voyage did less well.[11] documented A timekeeper as complex as H4 could not equip every ship, though. That took makers such as John Arnold and Thomas Earnshaw, whose simpler, reproducible designs had by the nineteenth century made the chronometer an essential instrument on naval and surveying vessels.[12] HMS Beagle carried 22 of them on its second voyage, from 1831, to carry a chain of longitude differences around the world.[12]
The Sun keeps uneven time
There is a catch in “find local time from the Sun.” The Sun is not a good clock. A sundial and a mechanical clock set to agree on one day will disagree by more than a quarter of an hour a few months later, then come back together. The difference is the equation of time. Drag along the year and watch the Sun’s noon position trace a figure eight, the analemma.
Two effects add up. Earth’s orbit is an ellipse, so Earth moves faster near perihelion in early January, and the Sun appears to move faster along the ecliptic then; that term swings by about ±7½ minutes over a year. Earth’s axis is tilted 23.4°, so even uniform motion along the ecliptic projects unevenly onto the equator, where clocks are kept; that term swings by about ±10 minutes twice a year.[13] Their sum peaks near +16½ minutes in early November and −14 minutes in mid-February.
For longitude this matters directly. If you time local noon by the Sun and treat it as 12:00 mean time, you are wrong by the equation of time, and every 4 minutes is a degree:
Longitude at noon, worked
The cleanest way to find local time from the Sun is to catch it on the meridian: local apparent noon, when it stands highest. If your chronometer says Greenwich time at that moment, the rest is arithmetic. The figure below simulates sextant sights taken every 2 minutes either side of noon by a ship really at 35° N, 40° W on 3 November 2026. Your task is to decide when noon happened.
The steps are: (1) read the chronometer at noon; (2) take the equation of time for the date, here about +16½ minutes; (3) local mean time at apparent noon is 12:00 minus that, 11:43:30; (4)
Now notice what the highest-sight method does. Near noon the Sun’s altitude barely changes: on this date it stays within half an arcminute of its peak for nearly 8 minutes. With sights scattering by half an arcminute, the highest one can land anywhere in that window, and 4 minutes of time is a full degree of longitude. Navigators knew this. Noon was the moment for latitude, not for time. To find time from the Sun they took sights when it was rising or falling fast, well away from the meridian.
The method of equal altitudes uses that. Take a sight in the morning and note the time; wait until the Sun comes down to the same altitude in the afternoon and note that time too. Halfway between is noon. Because the Sun is moving quickly at both sights, an error of half an arcminute in altitude becomes only a few seconds of time. A related method, the time sight, worked from a single altitude, the latitude and the Sun’s declination to compute local time by spherical trigonometry, and it was the everyday method of chronometer navigation through the nineteenth century.[14]
The Moon as a clock
The other road to Greenwich time runs through the sky. The Moon circles Earth in about 27⅓ days relative to the stars, so it moves about 13° a day, or a little over half a degree per hour, against the background. Its own disc is about half a degree across. In other words, the Moon moves about its own diameter every hour. If an almanac predicted where the Moon would be among the stars at each Greenwich hour, the sky would be a clock face, and the Moon its hand.
The quantity a navigator could measure was a lunar distance: the angle between the Moon’s bright limb and the Sun or a bright star near its path, taken with a sextant held on its side. In the figure, choose the Sun or Aldebaran and slide the time of the sight. The almanac page on the right lists predicted distances every 3 hours; find your cleared distance between two entries and read off Greenwich time.
The sensitivity is brutal. The distance changes by about 30′ per hour, so
That last requirement was the hardest. The Moon’s path is pulled about by the Sun and Earth’s oblateness; a useful theory needs dozens of periodic terms. Tobias Mayer, a cartographer and astronomer in Göttingen, produced lunar tables in the 1750s good enough for the purpose. James Bradley, the Astronomer Royal, judged that with Mayer’s tables and a good instrument, a ship’s longitude could be found within about a degree.[15] Nevil Maskelyne tested the method on his voyage to St Helena in 1761 and found it gave longitude within about 1½°, against dead-reckoning errors of up to 10°.[15] documented He published the procedure in The British Mariner’s Guide of 1763, and as Astronomer Royal he launched the Nautical Almanac, whose first volume, for the year 1767, was ready by the end of 1766 and listed lunar distances for every 3 hours.[16][15] documented The astronomy was not British alone. Mayer was German; the French astronomer Nicolas-Louis de Lacaille had earlier proposed tabulating lunar distances in advance; and the almanac was computed by a network of paid human computers working at home across England.[10]
Clearing the distance
The almanac could not list what you see. It listed distances between the centers of the Moon and the Sun as seen from the center of the Earth. What the sextant measures is a distance between limbs, seen from the deck, through the atmosphere. Converting one to the other was called clearing the distance, and it is the reason the lunar method was famous for its arithmetic.
Work down the right-hand panel. First add the two semidiameters, about 16′ each, to turn limb-to-limb into center-to-center. The Moon’s is enlarged slightly when it is high, because you are then up to 6,400 km closer to it than Earth’s center is. Next, each body has to be moved to where it would be seen from Earth’s center without air. Refraction has lifted both a little, by about 1′ at 45° altitude, more near the horizon. Parallax has pushed the Moon down: you stand about 6,400 km from Earth’s center and the Moon is only about 60 Earth radii away, so seen from the surface it sits lower in the sky by
These shifts move each body straight up or down, so they change the distance by amounts that depend on the angles of the triangle formed by the zenith and the two bodies. With apparent altitudes h′, true altitudes H, and the measured apparent distance d′, the cleared distance D follows from
In the default case the whole correction is about 20′, and skipping it would cost 40 minutes of time and 10° of longitude. Try lowering the Moon toward the horizon: parallax grows, refraction grows faster, and the correction changes quickly, which is one reason navigators preferred sights with both bodies well up. Navigators did not evaluate cosines to seven figures by hand at sea. Maskelyne’s Requisite Tables of 1767 and a long series of later methods and tables reduced the work to a sequence of table look-ups and additions; even so, a full lunar took a practiced navigator something like half an hour of careful arithmetic.[16][10] reconstructed
Clocks, Moon and radio
For about a century the two methods worked together. Chronometers were expensive and ships often carried one or none; lunars cost nothing but skill and an almanac, and they gave an independent check on a chronometer that might have stopped or jumped. A careful navigator with a good sextant could expect a lunar longitude within about half a degree, not much better than the Act’s strictest prize.[15][10] reconstructed As chronometers became cheaper, more reliable and more numerous during the nineteenth century, lunars faded into a backup skill. The British Nautical Almanac for 1907 left the lunar distance tables out altogether.[18] documented
The final change was to stop carrying Greenwich time at all and receive it. Telegraph time signals and time balls had long let ships check their chronometers in port. In the twentieth century radio brought the signal to sea: the United States Navy began regularly scheduled radio time broadcasts from the Boston Navy Yard on 9 August 1904, and the Eiffel Tower began broadcasting time from Paris on 23 May 1910.[19] documented With a daily time signal, a chronometer’s accumulated error no longer mattered; only the hours since the last signal did.
The chain of reasoning never changed, though. Every one of these methods found longitude by comparing local time, read from the sky, with reference time, carried, computed or received. The last chapters of this atlas come back to it: a GNSS receiver is, at heart, a clock comparison too. Before that, the next chapter opens the instrument that made both the lunar and the chronometer methods possible: How a sextant works.