We previously looked at how early explorers determined their latitude, but equally important for navigation is determining longitude. This can also be done by looking at the position of the sun, but it also requires precise and accurate timekeeping, which was such a challenge that the British government offered cash prizes to anybody who could accomplish this.

When determining latitude, it was important to use a sextant to determine the angle that the sun made against the horizon at its highest point. For determining longitude, it's important to determine the exact time when the sun is at its highest point.

The earth rotates about its axis one full rotation every day. We know that one full rotation is 360°, and this takes 24 hours. This means that the earth rotates 15° every hour. Every minute, then, the earth rotates 0.25° or 15'. And every second, the earth rotates at 0.25'. And this fact can actually be used to determine longitude!

The Greenwich Meridian is the name for the line of longitude at 0°, named so because it goes through the borough of Greenwich in London. To determine one's longitude, they need to know the following things:

1. The precise time that the sun was at its highest position in Greenwich.
2. The precise time that the sun was at its highest position at the navigator's position.

The thing that's important to note is that the clock that the navigator uses must be synchronized to clocks in Greenwich and not be changed to account for different time zones. This is key! And this was actually a huge difficulty in the late 19th and early 20th centuries because most clocks then relied on pendulums, and these pendulums would be thrown off when the clock was on a moving ship. Solving this problem is an entirely different story, though, and I want to focus on the actual calculations.

If you know the time that the sun was at its highest position in Greenwich, and you know the time that the sun was at its highest position wherever you're located, taking the difference of these times tells you how much the earth has rotated in this time. And this is equal to the difference in longitude between your location and the Greenwich Meridian, which is located at 0°.

So, for example, if an explorer was sailing around the world, and the sun reached its peak at 2:30:25 PM GMT (Greenwich Mean Time), and that day in Greenwich the sun reached its peak at 12:03:12 PM, this is a difference of 2:27:13. We know that the earth rotates 15° for each hour, 0.25° for each minute, and 0.25' for each second, so this gives us a total of 2*15° + 27*0.25° + 13*0.25' = 36°+6.75° + 6.75' = 36° + 6° + 0.75° + 6.75' = 42° + 45' + 6.75' = 42°51.75'. So that explorer was located at 42°51.75'W.

Note that if the time the sun reached its peak was earlier than in Greenwich, you would be east of Greenwich. The calculations would be the same, but then your longitude would be east instead of west.

Also note that it's difficult to do calculations down to the second in the real world. You'd need stability (hard to do on a ship), a very precise sextant, and someone who is very skilled at using it. And you'd need to know exactly when the sun reached its peak in Greenwich. However, this part is simpler because it can be looked up in an almanac (we know enough about the way the earth rotates around the sun that these things can be calculated precisely).

Sample Problems

(For these problems, use the chart titled Meridian Passage of the Sun 2020 in this almanac.)

1. Suppose you are exploring the world on January 1, 2020, and your sextant and chronometer show that at your position, the sun is highest at 4:00:03 PM GMT. What is your longitude?

2. Suppose you are exploring the world on March 15, 2020, and you your sextant and chronometer show that the sun is highest at 5:32:01 AM GMT. What is your longitude?

3. Suppose you are exploring the world on July 4, 2020, and your sextant and chronometer show that the sun is highest at 12:04:21 AM GMT. What is your longitude?