Time and place

The Longitude Game: Don't Get Lost at Sea

Goal: feel, rather than be told, that time difference is longitude, and that a small clock error becomes a large position error.

Two versions, same maths:

• Browser: site/longitude-game/: open index.html, or visit the

project's GitHub Pages site

• Terminal: moonfield longitude, no browser required

Neither is the "real" one. Use whichever suits you.

Observe

You are on a ship. There is no land in any direction. You have:

• A sextant, so you can find local noon and your latitude
• A chronometer, still showing the time at your home port
• No satellites, no radio, no landmarks

Where are you?

Predict

Before calculating: if your clock reads two hours ahead of your local noon, are you east or west of home? By how much?

Commit to an answer. Most people get the direction wrong first time.

Learn

The arithmetic

360 degrees / 24 hours   =  15 degrees per hour
15 degrees / 60 minutes  =  0.25 degrees per minute
1 degree                 ≈  4 minutes of time

The method

1. Watch the Sun. When it stops rising and starts falling, that is **local

apparent noon**; the Sun is on your meridian.

1. At that exact instant, read your home-port chronometer.
2. The difference is your longitude.

If your London chronometer reads 15:00 at your local noon, then noon reached you three hours after it reached London. Earth turns west to east, so the Sun arrives at eastern places first. Three hours later means 45 degrees west.

longitude = -(reference_time - local_noon) × 15

The minus sign is the part people get backwards. Your clock reading ahead of local noon means you are west.

Run

moonfield longitude --reference 14.5 --local-noon 12

  - Reference clock at local noon: 14:30
  - Local apparent noon:           12:00
  - Difference: +2.5000 hours

  Longitude = -(+2.5000 h) x 15 deg/h = -37.500 deg
  You are at 37.500 degrees west of your reference.

Change one variable: Wrong Landfall

Now break the clock.

moonfield longitude --reference 14.5 --local-noon 12 --drift 30
moonfield longitude --reference 14.5 --local-noon 12 --drift 120
moonfield longitude --reference 14.5 --local-noon 12 --drift 600

Thirty seconds of drift:

  - An error of 0.125 degrees
  - At latitude 45, that is about 9.8 km in the wrong place

Ten minutes:

  - An error of 2.500 degrees
  - At latitude 45, that is about 196.8 km in the wrong place

This is how ships were lost. Not through bad seamanship, but
through arithmetic done on a number that was quietly wrong.

The rule of thumb worth memorising:

4 seconds of clock error = 1 nautical mile of longitude error at the equator.

A chronometer losing one second per day is invisible on any given day, and thirty seconds (about 14 km) wrong after a month at sea.

Validate

Check the arithmetic yourself, on paper, for one case. Then check it against reality: your own timezone offset is roughly your longitude divided by 15. If you are at 74°W, you should be around UTC−5. You will find it is roughly right and not exactly, because timezone borders follow politics.

Explain

Why is latitude easy and longitude hard?

Because Earth's rotation gives latitude a natural reference (the equator and the poles are physically distinguishable) while every meridian is identical. There is no natural zero for longitude, so you must carry your reference with you. That is what the chronometer is: a reference you brought from home.

The generalisation is worth keeping:

A measurement is only as good as its reference, and errors in the reference propagate into the result multiplied by whatever constant relates them.

Here that constant is 15 degrees per hour. It is unforgiving, and it does not care that your clock error was small.

Checkpoint

• I can state the 15-degrees-per-hour relationship
• I can work out longitude from two clock readings
• I know which direction a clock reading ahead implies
• I can convert a clock error into a distance error
• I know why latitude was solved centuries before longitude
• I can explain why calibration matters, using this as the example

Try it yourself

1. Work out the longitude for a reference reading of 09:00 at local noon.

Which hemisphere?

1. How much clock drift puts you 1 km wrong at your own latitude?
2. Harrison's H4 lost about 5 seconds in 81 days. How far wrong was that

voyage, at the equator?

1. Compute the error a 1-second-per-day clock accumulates over a two-year

voyage.

Questions to think about

• The lunar-distance method needed no clock but hours of computation per fix.

Chronometers needed no computation but an expensive, fragile instrument. Which would you have backed in 1730?

• GPS works by comparing signal arrival times from satellites with known

positions. How is that the same problem, and how is it different?

• What is the modern equivalent of a drifting chronometer, a small, invisible

error in a reference that quietly poisons everything downstream?

Go deeper

• Why UTC exists
• Dava Sobel, Longitude (1995)
• The browser version in site/longitude-game/

Next: Module 02, Moon phases.