We've seen that it's pretty easy to determine your latitude using the sun as a reference point. All you need is a shadow and chart that was easily available to sailors of previous centuries and you're set. Finding your longitude is another story. The reason for the difference in difficulty is one of time. The sun varies very little in its north-south position from day to day, but it varies enormously in its east-west position by virtue of the whole "rising and setting" thing. It's easy to look up on a chart to find a correction factor that varies day-by-day, but when you need a correction factor that varies minute-by-minute you're in deep trouble especially if you live in the age before reliable clocks.
It was the work of John Harrison shortly before the American Revolution that first gave the world a clock accurate enough to navigate by. With a good clock you can determine to the second where the sun should be at a given longitude and thus reconstruct the longitude you're actually near.
Modern GPS uses almost this same method. All a GPS satellite does is eternally broadcast two continuously updated pieces of information: its position and the time on its atomic clock. Knowing that light travels at about 1 foot per nanosecond, we can calculate how far we are from the satellite to the foot, as long as the GPS clock is accurate to the nanosecond and we have a receiver that can handle such a precise signal. One satellite is not enough, because simply knowing a distance doesn't tell you direction. If you have two signals, you can eliminate some possibilities. Imagine I know how far I am from Houston and how far I am from College Station. In most cases this narrows down the possibilities to two, which we can see schematically for one hypothetical pair of distances:
I could be in Conroe or in a field north of Highway 90. One more position signal would eliminate the wrong possibility. In 3d we need four satellites, but in practice you'll be able to get signals from many more than that at a given time if you have a decent view of the sky.
All this relies on having a good clock. While a clock isn't usually thought of as a position-finding tool, we would be much the worse without them.
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Dava Sobel wrote book about John Harrison and the race for the Royal Society of London prize for a solid method of finding longitude at sea, called, of all things Longitude. It's short, but very good read, and is especially interesting in the battle between Harrison chronograph method and the lunar distance method.
Good stuff.
Also, even cooler, Harrison's actual cronographs are still functional and on display.
I'd nominate Harrison's H-4 as history's most significant timepiece.
Hopefully you said Highway 90 for the sake of your readers having to read a badly-labeled Google Map. Anyone who lives in Texas would call that I-10. *wink*
I think this explanation is skipping an important step. The GPS receiver can't determine directly the distance to the satellite based only on the time in the signal it receives. If it could, that would mean the receiver must contain an extremely accurate clock that was itself synced with the GPS satellites' clocks. But that's not true. What the receiver can measure, however, is the time difference between the arrival of signals from different satellites. From this the receiver can eventually determine distances to the satellites. I guess this is a small quibble. Otherwise I really like the post. GPS is a good example to show the use of math and geometry to make a really practical device.
Sobel's book about Harrison is fucking awesome!
It is important to understand that the GPS clock for position location has nothing to do with local time, although the GPS can easily calculate that. Local time is arbitrarily the same for approximately 15° intervals and calculating the exact sun time for a given position is essentially the same as determining its longitude.
This does not contradict the original posting, of course. It is the length of time between sending and receiving signals that determines the location circle.
The GPS system is also a continuous detector of the homogeneity of time throughout the local volume of space. If something naughty were to transpire, naturally or through instrumentality, the glitch would be detected as an impossible coordinate assignment.
Although Harrison successfully met the terms of the royal longitude challenge, his chronometers were far too complicated to manufacture in the quantity needed. (I recall something like 700 parts.) It took a succession of watchmakers many decades to simplify the design to the point of practicality. These simplification involved enormous ingenuity. The modern mechanical chronometer is a collaborative work, with no one deserving the majority of the credit. Harrison stands out for vision, versatility, persistence.
Another complication of the GPS system is the necessity to correct for general relativity: the receiver is deeper inside the Earth's gravity well than are the satellites.
Also worth noting is Differential GPS (DGPS) which is the use of ground stations with very accuratly known positions that broadcast error corrections. Originally made to correct for the "selective availibility" feature insisted on by the US military for some fairly ridiculous security reasons. This consisted of deliberatly screwing with the clocks to reduce accuracy to 100m for civilian users, and was only finaly turned off in 2000. The system remains in place for those who require very accurate position data as it eliminates errors caused by things like atmospheric effects.