The Longitude Problem

Figuring out how far north or south you are, even when at sea, is fairly easy. You just need to find the height of the sun at noon (i.e., when the sun is due north or south of your position) and apply a corrective factor based on the time of year. That is, all that's required to determine your latitude is a compass, a sextant, and a single table--all of which sailors would have had easy access to.

On the other hand, figuring out how far east or west you are is considerably more difficult, because of the rotation of the earth. If you want to determine your longitude, you have to contend with the fact that, if you're observing something in the sky now, someone several hundred miles to your east could have observed almost precisely the same thing an hour ago. Prior to a genuine solution, the general practice was to sail to the correct latitude then go East or West until you hit land, but that has certain drawbacks: not only is it remarkably inefficient, it makes it difficult to avoid any nautical hazards that are at the same latitude as your desired destination.

Historically, there have been two forms of attempted solution to this problem:

Observe celestial phenomena which vary rapidly. Generally the idea here was to make very precise observations of either the moon or the moons of Jupiter. Unfortunately, finding the moons of Jupiter while on the deck of a moving ship proved impractical. Observations using the moon, on the other hand, were possible in theory, but required extraordinarily complex computations in practice. Nevertheless, well-designed calculation tables made this the standard method for a while in the late 18th and early 19th century.
Have a very accurate clock. If you carry around a clock set to some fixed time, you can compare local time to time at that fixed location, and thereby determine how different your longitude is from its longitude. On the other hand, constructing a clock that keeps good time while on a moving ship is itself a very difficult problem. The first chronometer capable of keeping sufficiently accurate time while on a ship was made by John Harrison in 1761; for quite some time, nobody actually believed that a mere clock could solve the problem astronomers had struggled with for centuries, and even once people did listen they were prohibitively expensive for years, but eventually they became cheap enough to be the standard method of determining longitude.

Neither of these methods were viable until the middle of the 18th century; thus, any ships out of sight of land before that time would have had no good way of knowing where they were, leading to many navigational mishaps as well as wasted time at sea. For the British, this problem became intolerable and led to a massive 20,000 pound reward (on the order of ten million of today's dollars). The money was enough for an entire generation of bright minds to seek a solution.

In the modern-day era of GPS and radio signals broadcast by atomic clocks, this is almost never an issue.

For more details, see The Other Wiki, here.

Invocations of the longitude problem in media include:

Literature

The Baroque Cycle has a bunch of navigation without a solution to the problem. They need to keep their charts organized by longitude.
The Island of the Day Before by Umberto Eco contains a number of different attempts to solve the longitude problem, including one that uses Sympathetic Magic (the theory is that a wounded dog is taken on the ship; the sympathetic magic is performed on the dog every night at midnight in Paris; by watching the dog's reaction and noting the local time, you can figure out your longitude much as with the "clock" method).
In Devil of a Fix by Paliser Marcus, the MacGuffin is a secret almanac containing a solution to the longitude problem.
The Medici Emerald by Martin Woodhouse and Robert Ross has Leonardo da Vinci investigate the death of a friend of his, a merchant captain renowned for getting where he wanted to go at unheard-of speed. The captain had done a favor for an Arab mathematician who built a very accurate clock ... in the 1460s or '70s.
Shows up occasionally (as to be expected) in the Wooden Ships and Iron Men Aubrey-Maturin series of novels by Patrick O'Brian, usually when one of the ship's clocks has some kind of breakdown and makes navigation very difficult. Watches and clocks are rightly shown as being extremely important for any maritime venture, though not only for navigation.
- It was to the point that no ship captained by Aubrey carried only one chronometer onboard; mention is made of Jack consulting three to get the average. And he was still teaching midshipmen how to observe the moons of Jupiter throughout the series.
The focus of the book Longitude by Dava Sobel, who traces the various attempts to solve the problem but primarily focuses on clockmaker John Harrison.
- Adapted into a television mini-series staring Michael Gambon as Harrison, shown on A&E in North America, and Channel 4 in the UK.
The novel Mason & Dixon by Thomas Pynchon talks a lot about the Longitude Problem. This is reasonable when your protagonists are surveyor-astronomers tooling around the British Empire in the 18th century. Mason and Dixon test chronometers and work out moon-observation tables in the course of their careers.
One Time Machine novel (probably Sail With the Pirates)^{[please verify]} has the reader stranded on a small island in the Caribbean with some sailors. There's an argument when the sailors try to determine their latitude and the reader suggests figuring out their longitude. This is immediately shot down as impossible. Which is decidedly odd, because figuring a longitude of a fixed place, instead of moving ship, is relatively easy, if somewhat long process.
Gulliver's Travels is very strange about this. On the one hand, Gulliver carefully reports both the latitude and longitude of all the various fictional places he visits. On the other, when he's in Laputa fantasizing about what he could do if he were immortal, one of the problems he imagines being able to solve is "the discovery of the longitude". The effect is to create the impression that either Gulliver or Swift himself isn't entirely clear on what "the discovery of longitudes" actually means.
- As noted above, discovering longitude on land wasn't a major problem — sure, it required several careful astronomic observations and complicated calculations, but with some time it could be done very precisely. Sailors simply hadn't the time to do so, because their ship would certainly change its place between observations, and they needed a quick, even if somewhat less precise, way of figuring the longitude, instead of a very precise one that took a week to complete.

Live Action TV

The drama/documentary Longitude (see Literature above) is all about this problem; it mentions the "sympathetic magic" thing, together with other crackpot schemes like having dozens of ships anchored at fixed longitudes to provide reference points.
In the second episode of James Mays Man Lab, James tests the 'sympathetic magic' theory described above by attempting to sail to France using a dog as a navigational aid. Not surprisingly, it fails.

Tabletop Games

7th Sea incorporates this problem into the game setting. There's one country that's solved it - the country with teleportation magic among its nobility. Nobody's invented a clock that's accurate and robust enough to survive a sea voyage, but by bringing a noble aboard, having him teleport home, and then having him come back with the time, Montaigne has developed the most accurate charts in all of Theah.