Note that all of the above rely on celestial observations and time keeping.
It is no accident that we still use the time at the royal naval observatory in Greenwich England as “universal coordinated time” . (renamed from Greenwich Mean Time) That observatory’s reason for existence was to compile celestial sighting data into tables to be used for accurate navigation, thus everything navigation related, including time, was measured relative to that observatory’s location.
Actually, finding longitude was simpler than that. Put simply, one clock was set to Greenwich time, and the other was set to local noon, which the officers determined daily. By comparing the difference in time, you could figure out how many degrees east or west you were from the starting point (Greenwich). Obviously, this is dependent on fairly accurate clocks, otherwise you’ll be tens, if not hundreds of miles off after a while of not seeing land.
This is why GMT is the standard, and why the Prime Meridian runs through Greenwich.
Latitude was more easily found by shooting Polaris with the sextant. At the North Pole, the elevation of Polaris would be 90 degrees, and at the Equator, it would be 0 degrees (on the horizon). You could tell how far north or south you were by seeing how high Polaris was above the horizon. (hence 0 degrees and 90 for the equator and north pole respectively.)
Occasionally, pre-modern astronomers would use simultaneous observations of eclipses. If you could tell when a particular phase of the eclipse occurred according to your local time, and your distant colleague eventually sent word of when the same phase had occurred according to his own local time, you could figure out how far apart you were in longitude. It was generally rather tough to orchestrate, though, and there were few reliable records of such observations before modern times.
Have you actually done this? How do you shoot Polaris with a sextant when it’s too dark to see the horizon?
Just one more vote that the OP reads Sobel’s book. It’s superb.
Just to add that the Old Royal Observatory in Greenwich is now a small museum. Last time I was there (probably 2004) it had an excellent display on the search for longitude, including some of Harrison’s chronometers.
If anyone interested in this subject finds themselves in London, it’s well worth the hour or so it takes to look around.
On a related point, while dealing with the recent switch from daylight saving time in the USA, I started thinking about the era before time zones, when local time was truly local. How did they accurately determine when the Sun was at its zenith (local noon)? I’m assuming that you would need that information plus a correction from the equation of time for seasonal variation from the mean solar day to accurately set a local clock.
Well since I missed the opportunity to recommend Longitude by quite a bit, I’ll throw out one for Revolution in Time, which covers the entire history of clocks and time measurement, including the many different ways people used to determine local time, with varying degrees of accuracy.
No, the method of determining longitude did not inevitably mean that Greenwich was adopted as the prime meridian. Britain adopted it, but other European countries adopted other prime meridians for their own purposes. For example, the French used a prime meridian running through Paris.
That of course created a great deal of confusion, since there was no universal system of longitude, unlike the universal system of latitude. Eventually, about a century after Harrison developed a reliable chronometer, there was an international convention in 1884 by which countires agreed to adopt the Greenwich meridian as the Prime Meridian. Britain and the U.S. both favoured it, and since they together represented a substantial portion of international shipping, other countries were persuaded to go along. Even so, France abstained from the final vote, and did not adopt Greenwich as the Prime Meridian until 1911.
Wiki link: International Meridian Convention
Lewis and Clark used the lunar eclipse method at one point on their journey, in the Mandan Village in what is now South Dakota. It works fine, but lunar eclipses aren’t all that common, so they were only able to use it for that one point.
Eclipses of the moons of Jupiter are much more common, and can be used, too, but those have their own problems. First, you need better instruments to see them (at least a telescope, compared to the naked eye for our own moon). Second, Jupiter is far enough away that the speed of light becomes an issue: The first half-decent measurement of the speed of light (better than Aristotle’s result of “really, really fast”) came from an astronomer who was trying to track down errors in timing data of Jupiter’s moons.
And sighting the horizon works well at sea, but on land the horizon is typically uneven enough to be useless for navigation. What you use instead is an “artificial horizon”: You have a pool of water or other liquid in a glass enclosure (to shield it from winds), and take a sighting between the object you’re interested in and its reflection in the water. Divide that angle by two, and you get the height above the true, flat horizon.
I didn’t mean to imply that there was anything special about Greenwich, other than it was where the Royal Observatory was, and therefore where the British master clock was.
It could have just as easily been the meridian through Paris, like you say, or even New York. Any meridian is as good as another for navigation, but ultimately, Greenwich became the “Prime Meridian” and 0 degrees because of British maritime dominance (and I’m guessing due to the charts, etc… being published for them)
As for the latitude, there was another way of determining it via the sun. The Polaris one is just the most obvious one, that’s all.
The French prime meridian was the Rose Meridian, if I remember correctly, and has a major starring role in the Da Vinci Code. 
A micro-nitpick …
“UTC” stands for Coordinated Universal Time, not “universal coordinated time”. At the international convention which created it, they couldn’t decide between the English acronym “CUT” & the French acronym “TUC”. So they compromised with one that is sorta-right & sorta wrong in both languages.
ref Coordinated Universal Time - Wikipedia (and other legit sources).
ETA: that article is really a good introduction to modern serious timekeeping & all the hairy details that flow from it.
indeed. 
For the record, Dan Brown was badly confused about Paris meridians. He mixes up the fact that there was a medidian line installed in St Sulplice on the Left Bank with the much more famous Paris Medidian that runs through the Observatory, about a mile or two south. The E-W difference between the two is only a few hundred feet, but that’s far more than can be attributed to experimental error.
The medidian in St Sulplice isn’t particularly unusual - there are quite a few European churches with one. John Heilbron’s The Sun in the Church (Harvard, 1999) is both definitive and highly recommended as an account of how such installations represent the way Catholic astronomers responded to the restrictions on them after the Galileo affair.
[The puzzle about the Paris meridian that’s intrigued Angua and myself recently is why it’s visible in the grass in the Google Earth image of Parc Montsouris, about another mile south of the Observatory. No, this is nothing to do with the Mire du Sud in the park, which isn’t quite correctly aligned anyway. We suspect that there was a N-S path once-upon-a-time laid out in alignment with the medidian in the park and since covered over, but we haven’t found proof.]
Montsouris is one of my favorite parks in Paris!
This page gives a brief description of the old Paris meridian’s North (mire du nord) and South (mire du sud) bearing points. If one takes the line of Arago medallions as being the true Paris meridian, then the mire du nord (in Montmartre) is several meters too far west, and the mire du sud several meters too far east.
There’s a good description of the Arago medallions here, containing a link to a Google Earth overlay showing the locations. From this, it’s clear that the rough part of the grass in the Parc Montsouris is exactly on the Arago line.
Now, there are several features in Paris that are nearly aligned to the Paris meridian; the long axis of the Jardin du Luxembourg is tantalizingly close, and the above-mentioned mires are even closer. It seems fair to assume that the orientation errors were due to limitations of contemporary instrumentation. That the “Montsouris grass-path” lines up exactly with the Arago medallions (placed in 1994-5) suggests to me that perhaps it’s a matter of cause and effect.
There seems to be significant interest in “walking the meridian”: here, here, and of course Dan Brown’s own protagonist Robert Langdon. However, it’s nowhere near as easy as it’s made to seem in The Da Vinci Code; markers are rarely close together or on a continuous street. Where the meridian does cross open land, it’s usually either hard ground (Louvre courtyard, Luxembourg pathways) or inaccessible to the public (Observatory grounds, Luxembourg formal flower-beds).
However, there’s one location where there are nine Arago medallions, and the line between them runs across an informal grass lawn (which is fun to walk on)… Parc Montsouris!
So, my guess is that since the mid-90’s, enough people have gone medallion-bagging across the lawn in Montsouris that the results are visible in Google Earth.
[Perhaps you and the lovely Angua could revisit the Parc some time soon and see if this theory holds water? Just in the interest of science, of course… I’d offer to do it myself but it’s a bit far.]
Dan Brown fail Geography? That’s unpossible !
Although Dan Brown’s Paris is often completely wrong, he is at least an equal opportunist; he also managed to put Rosslyn Chapel and Glastonbury “precisely” on the same meridian (that’s about a 17-mile, or ~5%, error).
[Of course, since it’s a novel, none of this would matter one whit if he weren’t quite so insufferable about his supposed “accuracy”.]
On a clear night the horizon is where the stars end.
Doesn’t work. Stars are always hard to see near the horizon (because too much atmosphere intervenes).
Nice! We’d been fiddling around, scrolling back and forth and the like, but that overlay nicely confirms the observation.
In the case of the Jardins du Luxembourg, the layout of the gardens predates the Observatory. The cause and effect here is that the latter was built to align with that axis, though since this was merely a grand bit of urban planning the alignment only had to be good enough to make the architectural point. While the meridian instrument in the Observatory was built into the design of the building, it was only over time that the notion developed that this defined a geographical meridian extending across the city, France and around the globe.
The mires were only ever intended as symbolic and the errors in their placement aren’t because of any unavoidable observational errors. Much better accuracy had been attained over those stretches by Delambre and Méchain.
It’s an interesting hypothesis and not one we’d thought of. My immediate reservation about it is that I’m not sure that “medallion-bagging” is that popular. I certainly take note of them if I notice them in the street, but I’ve never noticed anyone else look at one. There is a book published (in French) on walking the meridian that’s reasonably easy to find in Paris bookshops, but it’s not sufficiently common to suggest that it sells that many copies. We’ve taken the opportunity to detour through the park specifically to look at the mire and we didn’t bother following the medallions across the grass.
The other hypothesis someone suggested was that it was traces of the Bastille Day 2000 party: the city set up trestle tables along the route and invited people to bring picnics as part of this being done along the meridian all across the country to mark the Millennium. I happened to be in the city that day - and was rather disappointed that they hadn’t been more rigorous about laying out the tables, going instead for the easy option of following the streetplan so as to roughly follow the actual line. I wasn’t down to Montsouris that day, so don’t know how strict they were in the park, but it’s possible they laid the tables out more exactly there. Whether that would have been enough for the pattern to show up in the grass years later seems more dubious.
We certainly intend to take a closer look at the grass whenever we’re next in the park. However, any traces of whatever seem more likely to show up when the ground is dry, so we’re unlikely to go out of our way specially until next summer.
Awesome replies, everyone! Somehow, I never heard of the “sea-going chronometer” problems, and such. I WAG one problem may be how the ship’s own motion would affect any mechanical clock.
Thanks to all for their replies, and esp for the links to more info. I’ll have to read-up on this!