For centuries, explorers and sailors have used the Sun, in the absence of distinguishing landmarks, to find their way. Before we discuss how that works, a short background on navigation in general.
It's very important to know where you are when you're sailing. There are hazards under water: rocks, reefs, sand bars and wrecks. And there's the biggest hazard at sea - land. In daylight, with good visibility and in familiar waters, all you need is your local knowledge. And, although you never should be without one - you could manage without a chart. However, under the same conditions in unknown waters you will need a chart. You can match up the terrain with the chart and get a fairly good idea of where you are. A compass will allow you to determine your heading (the direction in which you are moving). You can also use a compass, or similar instruments to take the bearings (the direction, usually in degrees, from your position to the landmark or features in the landscape). If the skipper can see a familiar grain silo to the southwest, he or she knows the boat is somewhere on a line drawn through that feature, from northeast to southwest. This line is called a line of position. By taking the bearings of two suitable objects at about the same time, you can pinpoint your position. You are at the point where the two lines cross. You then know exactly where you are. It's also possible to use the same landscape feature twice. If you take one bearing and then run a straight course for a known distance and heading and make a second sighting, you can do a little simple trigonometry and determine your position. This is called a running fix.
There are many other ways of getting two lines of position. For example, if you can identify a lighthouse and find it listed in the table of lights, you will know its height. A little bit of high school mathematics will tell you how far away it is. Knowing that, you can draw a circle, centred on the lighthouse.
Knowing your speed and direction, you can safely leave the helm to brew a pot of tea without running aground.
The Need for Celestial Navigation
But what if you are sailing the Atlantic? There are no church steeples or lighthouses from which to take bearings. Instead, as a conscientious skipper you will use a technique called dead reckoning. You will have faithfully marked every course change on the chart. You will have plotted your progress using your compass heading and speed through the water.
You think you are still a good day's sail from the rocky Canadian coastline. Can you really go to sleep and be sure you won't crash on the rocks overnight? What if there were currents or tides that added to your speed through the water and you are miles ahead of your dead reckoning position? What if you've been close hauled1 for days and have been moving sideways through the water and you're going to make land fall at a different part of the coast?
An Accurate Way of Determining your Position.
As was pointed out in Using the Sun to Orient Yourself you can estimate where south is by the position of the Sun, knowing the time of day. This is fine if you are on a day's hike in the woods and you just need to know where south is, so you can hike to some point on the east/west highway. But the ocean sailor needs much more precision.
The Sun's Geographical Position
As you read these words, somewhere on our planet the Sun is exactly overhead. This point is called the Geographical Position. The point where that occurs is moving over the surface of the Earth, from east to west at one mile every four seconds. It's also moving very slowly north/south, taking a year to go from the Tropic of Capricorn to the Tropic of Cancer and back again, crossing the Equator twice in the process.
Someone has produced tables (they are published in the Nautical Almanac) that list minute-by-minute the geographical position of the Sun.
The lucky sailor who...
Is exactly under the Sun
Can verify this by measuring the 'altitude' of the Sun
has a watch that keeps very good time
has the tables that list where (longitude and latitude)2 the Sun is exactly overhead at any given time and day
... should know where he is located.
Now, a sailor is unlikely to be at that exact location. So the Sun will not be exactly overhead. Imagine a group of observers directly under the Sun. If they disperse to form a circle they will all see the Sun at the same angle: instead of 90°, maybe at 80°. This angle is called the true altitude.
A nautical mile is bit larger than a land mile. Its length was not chosen at random but rather, it is the distance on the surface of the Earth subtended by an angle of one minute (one sixtieth of a degree) at the centre of the Earth. If our observers backed away 60 nautical miles from the geographical position of the Sun, the Sun would drop one degree in the sky.
In other words, if you measure the altitude of the Sun you can find out how far away from the Sun's geographical position you are. You could look up in the tables the geographical position of the Sun for the date and time of your observation and draw a huge circle on the chart. You are somewhere on that circle. Unfortunately, that line of position could be thousands of miles in diameter. We need some more information.
Tabulated Altitude and Azimuth
You have your rough dead reckoning position. From that you can calculate, or look up3 what the Sun's altitude should be at that position. You can now draw a second, concentric circle. It may be inside or outside of the first one.
If we know the latitude and longitude of two places on the Earth, we can calculate how far apart they are and the direction from one to the other. The direction from the dead reckoning position to the Sun's geographical position is called the azimuth.
The InterceptDraw a line from the dead reckoning position towards or away from the Sun's geographical position (depending on where the dead reckoning position lies with respect to the first circle) using the azimuth. Where this crosses the first circle you drew is almost, but not exactly your true position.
In practice you wouldn't draw the entire first circle. The centre might be in Africa. You may not even have a map of Africa with you if you are sailing the Atlantic. A short straight line is an acceptable approximation. The second circle doesn't have to be drawn at all.
You would actually proceed as follows:
Measure the true altitude of the Sun.
Determine the tabulated altitude of your DR position.
Take the difference between them (in minutes, which is also nautical miles). Let's say it is eight miles.
Plot your dead reckoning position on the chart.
Draw a line towards (or away from) the Sun's geographical position using the azimuth from the DR position.
Measure off eight miles, or whatever your difference was.
At this point draw a line at right angles. This is a small part of the boat's position circle
The Sun-run-Sun Sight
The short part of the position circle we drew does not yet give us our exact position. But just as in terrestrial navigation where we can take two sights from the same object, we can make a Sun-run-Sun sight. The final position will still be subject to errors: it will be as accurate as the course steered and the distance run. It should not be a run of several days, but rather just a few hours.
Thus far we have assumed that we can measure the altitude of the Sun. However, there is no convenient marker to show us where the zenith (the point in the sky that is exactly overhead) is. It would be difficult to measure the angle between this imaginary point and the Sun. But, at 90° to the zenith is the horizon. We measure the height of the Sun above the horizon and take the complement instead.
To do this we use an instrument called a sextant. It has a system of mirrors and shades that allow us to view an image of the Sun and bring it precisely to the horizon. We read the exact angle off of a scale, and a micrometer.
The sextant is a delicate instrument. It must be cared for, set up and calibrated. Its known errors must be considered when using the results to look up tables.
Because the Sun appears to be a fairly large disk, we do not attempt to guess where the centre is. We bring the bottom edge of the Sun to the horizon. There is an adjustment factor, the semi-diameter correction, for both the Sun and the Moon. Stars appears as very small points and the factor does not apply.
The height of the observer's eye above sea level is another factor to be taken into account. It is expressed as an angle, dip, but can be looked up in a table, given the height in metres.
Ideally, we should be at the centre of the Earth when taking the sighting. Otherwise we have a parallax error. This factor is least when the Sun is directly overhead and becomes more of a factor the lower the Sun is. A sight would normally not be taken when the Sun is lower than 15°. Again, there is a look up table to determine the correction factor.
Refraction is also an issue. Like parallax, it's a function of how low or high in the sky the Sun is. And, like parallax, you need to make sure to take it into account.
Three of the above (semi-diameter, parallax and refraction) are combined into one adjustment.
With the geographical point moving so quickly, the exact time of day when the sight is taken is obviously critical. An accurate chronometer is essential. Ideally, there would be a master chronometer, safely below deck. Another watch is taken above to make the sight. Knowing the difference between the two we can deduce the real time of the observation. In turn, the master clock can be verified against a radio signal.
The Noon Sighting
There is something special about a sighting taken at precisely noon. That's not Greenwich Mean Time noon, or even noon by your watch. It is noon Local Mean Time: The natural noon for your meridian or longitude. It is of course, the time when the Sun is at its highest. But to know the LMT, we need to know our longitude, a catch-22 situation. But again, our dead reckoning position gives us the starting point. In the Nautical Almanac we can look up the time of high noon for our rough position. This is an estimated time. We start taking sights before that time. We take a sequence of readings, looking for the highest. The Sun's declination (from the almanac) combined (algebraically) with the complement of the reading we just took is our latitude. Or, expressed another way: if we were to plot a small part of the position circle as a straight line on the chart, it would be an east-west line.
Other Heavenly Bodies
Just like the Sun, any other fixed heavenly body will have a predictable geographical position. The Almanac lists tables for our Moon, Venus, Mars, Jupiter and Saturn and some 57 stars. Planets are best sighted at twilight, when the horizon can still be made out. If the horizon is not visible, then the sighting of two stars is needed. This can be an advantage. We can take two, almost simultaneous sightings using two objects and get an exact position without having to do the equivalent of the Sun-run-Sun fix.
There are complications. We rotate under the Sun at 15° every hour. But unlike the Sun, the planets are also moving, relatively quickly on their orbits. There's a correction for this. It's always positive, except, sometimes4, for Venus and is applied to the Greenwich Hour Angle.
What about GPS?
You may be asking about GPS. It is true that the Global Positioning System means that anyone can find their exact position with a simple, inexpensive, hand-held electronic unit. But electronics can fail. There is still a good argument to be made for learning how to navigate the old-fashioned way. It's still part of the syllabus for most countries' Yachtmaster certification.