Horology · navigation · the arithmetic of a prize
Wrong the Same Way Every Day
The story everyone knows is that John Harrison built an accurate clock and won the Longitude Prize. Read the 1714 Act closely and something stranger falls out. The prize never asked for accuracy. What it demanded, once you turn its words into arithmetic, is a clock that is wrong the same way every day: steady enough that whatever it loses, it loses on a fixed schedule you can subtract back out. A watch running ten minutes fast finds land perfectly. A watch dead-on today and wandering tomorrow is lost at sea.
This is the same move the Artificial Wasteland keeps finding under ordinary nouns: the thing that matters is not the level but the rate. Here the level is a red herring the Board of Longitude could see through, and the rate is the whole game.
First, the exchange rate: longitude is time
Everything below rests on one fact, and it is exact by definition. The Earth turns once, 360 degrees, in a day. So each degree of longitude is 240 seconds of clock time (86,400 ÷ 360), and each arc-minute of longitude, one geographical mile in the Act's language, is exactly four seconds. Know Greenwich time to four seconds and, at the equator, you know your longitude to one nautical mile. The whole problem of longitude at sea is the problem of carrying an accurate Greenwich clock across an ocean, on a wet, pitching, temperature-swinging deck, for weeks.
The 1714 Act, An Act for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea, set three tiers, and its own words are metric, not vague. The reward was £10,000 "if it determines the said longitude to one degree of a great circle, or sixty geographical miles", £15,000 "to two thirds of that distance", and the full £20,000 "to one half of the same distance". Read those distances as time budgets, and then divide by the length of the voyage, and each prize becomes a demand for a daily rate.
| Prize | Within | = time budget | ÷ 42-day crossing |
|---|---|---|---|
| £20,000 | ½° · 30 nm | 120 s | 2.86 s / day |
| £15,000 | ⅔° · 40 nm | 160 s | 3.81 s / day |
| £10,000 | 1° · 60 nm | 240 s | 5.71 s / day |
Time budgets at the equator, where a nautical mile of longitude is 4 s of time. The daily figure is the budget spread evenly across a six-week passage. A "six-week" crossing is a working average for the period, not a term of the Act; drag the voyage length below to see how the demand tightens or loosens with it.
Two-point-eight-six seconds a day. That is the real specification behind the most famous prize in the history of engineering: build something that, out of the 86,400 seconds in a day, mis-keeps fewer than three of them and does it the same way tomorrow. Because here is the turn that the whole thing hinges on.
You do not need a clock that keeps perfect time. You need a clock whose error is predictable.
Before a chronometer sails, it is compared against a known clock for days, and its going rate, how many seconds it gains or loses each day, is measured and written down. At sea you do not read the dial; you read the dial and then correct it by the rate times the days elapsed. So a watch that loses a steady 3 seconds a day is as good as a perfect one. What you cannot correct is the part of the rate that changes: the day it loses 3, then the day it loses 5, then the day it gains 1. That wander, the instability of the rate, is the only error that reaches the landfall.
The instrument: sail it yourself
Set a clock's average rate and its instability, then sail from Britain to the West Indies and see where the ship thinks it is. The average rate is subtracted out perfectly, so slide it as far as you like: a clock losing a steady minute a day still lands you on the dock. It is the instability, the random jitter added to the rate each day, that scatters the landfall. The rings are the three prize thresholds around the destination port.
Illustrative Monte Carlo, not a claim about a specific voyage: each crossing draws an independent daily rate error from a normal distribution with the instability you set, accumulates it over the voyage, and converts the leftover seconds to a landfall offset (4 s = 1 nm of longitude, scaled by cos of the destination latitude). The average rate cancels exactly, which is the whole point; only the instability survives. Prize rings are the Act's real thresholds.
Two voyages that settled it
Harrison's fourth timekeeper, the watch called H4, went to sea twice under the Act. Both times the number that mattered was not how far off the dial read, it was how little the rate had wandered. Here is what each trial actually recorded, converted into the currency of the prize.
- Rate set at Portsmouth
- −2.67 s/day
- Voyage measured
- 81 d 5 h
- Residual error on landfall
- 5 s
- = longitude
- 1.25′ ≈ 1 nm
- Unmodelled rate drift
- 0.06 s/day
- vs the £20,000 budget
- ≈ 25× inside
- Outbound voyage
- 46 days
- Residual error on landfall
- 39 s
- = longitude at 13°N
- ≈ 9.5 nm
- Unmodelled rate drift
- 0.85 s/day
- vs the £20,000 budget
- ≈ 3× inside
- Observer
- N. Maskelyne
On the Jamaica run, H4's landfall error was one nautical mile. The top prize allowed thirty. The watch had beaten the most stringent tier of the Act by a factor its makers could hardly have dared claim in advance, and it did it not by being magically accurate but by drifting from its set rate by six hundredths of a second per day. On the Barbados run, watched by Nevil Maskelyne, who had his own reasons to want the clock to fail and his own rival method to promote, it still landed inside ten miles, comfortably inside the £20,000 threshold again.
The Board of Longitude, unpersuaded that any clock could be this good and not merely lucky, awarded Harrison half the money, extracted his design in full public disclosure, and demanded he build more copies to prove the trick could be repeated by other hands. He got the balance in 1773, at the age of eighty, after the King intervened, and never a single line of it as "the prize" formally awarded. The steadiness was real; the institution took a decade longer to believe it than the arithmetic did.
The alternative it beat: reading the Moon
There was a rival answer to longitude that needed no fragile machine at all: the Moon itself is a clock. It moves against the background stars at about half a degree an hour, its own width in an hour, so the angle between the Moon and a chosen star tells you Greenwich time if you carry tables that predict where the Moon will be. Maskelyne championed this lunar-distance method, and from 1767 the Nautical Almanac published the tables that made it practical. It stayed in print for the method until 1906.
But look at the error budget and you see why the clock won in the end. The Moon moves so slowly that a measurement good to half an arc-minute, near the practical floor of a sextant in trained hands, already costs you about a minute of Greenwich time, and so about a quarter-degree of longitude: roughly fifteen nautical miles at the equator, on a good sight in calm conditions. That is the best case of the lunar method sitting near the worst acceptable tier of the clock. The lunar distance was cheap, needed no winding, and could recover from a smashed instrument; the chronometer, once it was cheap enough to carry two or three as a check on each other, was simply more precise and needed no clear sky. For a century ships carried both, and the Moon was the backup that saved you when the clocks disagreed.
The check
Every load-bearing number here, the prize tiers and their seconds-per-day, the four-seconds-per-mile exchange rate, both H4 residuals and their rate drifts, the lunar-distance ceiling, and the Almanac's dates, is re-derived from the constants and cross-checked against the trial records in a verifier. It also reads this page's own text and asserts the printed numbers, so a silent copy-edit that changed one would turn it red.
Run it, and what each figure rests on
From a fresh checkout: node verify-wrong-the-same-way-every-day.mjs. The arithmetic lives in research/wrong-the-same-way-every-day/compute.mjs; every cited number, with its source, is in research/wrong-the-same-way-every-day/facts.json and SOURCES.md.
| Claim | Rests on |
|---|---|
| 1° longitude = 240 s; 1 nm = 4 s | Definitional: 86,400 s ÷ 360°. Exact. |
| Three prize tiers & their words | Longitude Act 1714 (13 Ann. c. 14), quoted verbatim via The British Longitude Act Reconsidered, American Scientist. |
| £20,000 → 2.86 s/day over 42 d | 30 nm × 4 s ÷ 42 d, computed in compute.mjs. |
| Jamaica: 81 d 5 h, rate −24⁄9 s/day, 5 s / 1.25′ / ≈1 nm residual | Trial record via Wikipedia, John Harrison. Drift 5 s ÷ 81.2 d = 0.062 s/day. |
| Barbados: 46 d outbound, 39 s / <10 nm residual, Maskelyne observing | Wikipedia, John Harrison; Royal Museums Greenwich, Longitude found. Drift 39 s ÷ 46 d = 0.85 s/day. |
| Lunar distance ≈ 15 nm at best; Moon ~30′/hr | Wikipedia, Lunar distance (navigation). |
| Nautical Almanac tables 1767–1906 | Wikipedia, Lunar distance (navigation). |
Seams named honestly. The 42-day crossing is a period average, not an Act term; the Act names a West-Indies voyage but no fixed duration, so the daily figures scale with whatever passage you choose. The Barbados residual is reported as 39 s ("less than 10 miles") here; some accounts give ~54 s over a longer accounting window, and the difference is which leg you measure. The landfall Monte Carlo is illustrative of the level/rate distinction, not a reconstruction of any specific voyage's weather or track: it assumes the error is predominantly east-west and small, so a longitude arc scaled by cos(latitude) stands in for the great-circle miss.
The Longitude Prize is usually told as a triumph of accuracy over the establishment. It is at least as much a triumph of a subtler idea: that a wrong measurement is useful, sometimes more useful than a right one, if it is wrong reliably. A steady error is not an error at all once you know its size. That is the quiet thing a marine chronometer sells you, and it is why the number the Board should have cared about was never how far the dial was off, but how little that offset changed from one day at sea to the next.