(5a) Navigation

Lesson Plan for (5a) Navigation

I must go down to the sea again
To the lonely sea and sky
And all I ask is a tall ship
And a star to steer her by
Sea Fever by John Masefield

How does a captain determine a ship's position in mid-ocean? In our space age, this is easily done, by using the GPS system of satellites--the Global Positioning System. That network of 24 satellites constantly broadcasts its positions, and small hand-held receivers exist which convert those signals into positions accurate within 15 meters or about 50 feet.

Before the space age, however, it was not as easy. One had to use the Sun and the stars.

Finding latitude with the Pole Star

Imagine yourself standing at night at point P on Earth and observing the pole star (or better, the position of the north celestial pole, near that star), at an elevation angle h above the horizon.

The angle between the direction of the pole and the zenith is then (90°-h) degrees. If you continue the line from zenith downwards (see drawing) it reaches the center of the Earth, and the angle beween it and the Earth's axis is also (90°-h).

Therefore (as the drawing shows) h is also your latitude.

 The angle l of the pole star
 above the horizon equals the
local latitude

Finding latitude with the noontime Sun

If you are sailing a ship in mid-ocean, you can get the same information from the noontime Sun--probably more accurately, since at night you might not see the horizon very well.

Noon is when the Sun reaches the highest point in its journey across the sky. It then crosses the north-south direction--in the northern hemisphere, usually south of the observer. Because the axis of the Earth is inclined by an angle e = 23.5° to a line perpendicular to the ecliptic, the height of that point above the horizon depends on the season. Suppose you are at point P. We examine 3 possibilities:

 Position of the noon Sun
at the winter solstice
(1)   Suppose the date is the winter solstice, around December 21, when the north pole is inclined away from the Sun. To find your latitude l you measure the angle a between the direction of the noontime Sun and the zenith.

Look at the drawing and imagine you could rotate
the equator and the north pole N
until they reached the
   ecliptic and the pole of the ecliptic N'.
Then all three angles marked e fold up together, showing that they are equal. You get

a = l + e
and your latitude is

l = a - e = a - 23.5°

(2)    Half a year later, at the summer solstice (June 21), the north pole is inclined towards the Sun, not away from it, and now (if l is larger than e)

a = l - e
and your latitude is

l = a + e = a + 23.5°

 Position of the noon Sun
at the summer solstice
Position of the noon
Sun at equinox
(3)   Finally, suppose you are at equinox, around March 21 or September 21. The inclination of the Earth's axis is now out of the plane of the drawing--away from the paper, if this were a picture in a book. The direction to the Sun is in the plane of the equator, and we get

l = a

Thus at least at those dates, seafarers could tell what their latitude was by measuring the position of the noontime Sun.

For any other date, navigation tables exist that give the proper angle (smaller than 23.5 degrees) which must be added or subtracted. They also provide formulas for deriving the height of the noontime Sun from observations made at other times.

As with the pole star, rather than measuring the angle a from the zenith--which is not marked in the sky!--it is simpler to measure the angle (90°-a) from the horizon, which at sea is usually sharply defined. Such observations, known as "shooting the Sun," are done with an instrument known as the sextant. It has a sliding scale covering 1/6 of a circle (hence the name) and an attached pivoted mirror, providing a split view: by moving the scale, the sea-officer brings Sun and horizon simultaneously into view and then reads off the angle between them.


In the age of the great navigators--of Columbus, Magellan, Drake, Frobisher, Bering and others--finding your latitude was the easy part. Captains knew how to use the noontime Sun, and before the sextant was invented, a less precise instrument known as the cross-staff was widely used.

Longitude was a much harder nut to crack. In principle, all one needs is an accurate clock, set to Greenwich time. When the Sun "passes the meridian" at noon, we only need to check the clock: if Greenwich time is 9 a.m., we know that 3 hours hence it will be noon at Greenwich and we are therefore at longitude 15° x 3 = 45 degrees west.

However, accurate clocks require a fairly sophisticated technology. Pendulum clocks can keep time quite accurately on firm land, but the pitching and rolling of a ship makes them quite unsuitable for sea duty.

Non-pendulum clocks--e.g. wristwatches, before they became electronic--use a balance wheel, a small flywheel rotating back and forth through a small angle. A flat spiral spring is wrapped around its axis and it always brings the wheel back to its original position. The period of each back-and-forth oscillation is then only determined by the strength of the spring and the mass of the wheel, and it can replace the swing of the pendulum in controlling the motion of the clock's hands.

Gravity plays no role here, and motions of the ship also have very little effect; as discussed in a later section, a vaguely similar method was used in 1973 for "weighing" astronauts in the weightless environment of a space station. For navigation, however, such a clock must be very accurate, which is not easy to achieve: friction must be minimal, and so must changes in the dimensions of the balance wheel and properties of the spring due to changing temperature and other factors.

In the 17th and 18th century, when the navies of Britain, Spain, France and Holland all tried to dominate the seas, the "problem of longitude" assumed great strategic importance and occupied some of the best scientific minds. In 1714 Britain announced a prize of 20,000 pounds--a huge sum in those days--for a reliable solution, and John Harrison, a British clockmaker, spent decades trying to achieve it. His first two "chronometers," of 1735 and 1739, though accurate, were bulky and delicate pieces of machinery; they have been restored and are ticking away on public display, at the Royal Astronomical Observatory in Greenwich. Only his 4th instrument, tested in 1761, proved satisfactory, and it took some additional years before he received his prize.

A recommended source is the book Longitude by Dava Sobel.


Once radio arrived on the scene, early in the 20th century, the accuracy of chronometers became less critical, because broadcast time signals allowed shipboard timepieces to be reset periodically. But until then chronometers were essential to accurate navigation, as the following story illustrates.

In 1893 the Norwegian explorer Fridtjof Nansen set out towards the north pole (located in the ice-covered Arctic Ocean) in a specially strengthened ship, the "Fram." Having studied the currents of the Arctic Ocean, Nansen allowed "Fram" to be frozen into the polar ice, with which it slowly drifted across the water. Nearly two years, later, realizing that the course of "Fram" fell short of the pole, Nansen (who had prepared for this possibility) left the ship with his colleague Johansen and attempted to reach the pole by sleds over the ice. About 400 miles short of the pole they had to turn back: they wintered on a desolate island, in a hut they built of stones and walrus hides, and the following spring they headed for the islands of Svalbard (Spitzbergen).

They had been in the icy wilderness for more than a year, completely out of touch, but they always knew exactly where they were, because each man carried a spring-powered chronometer. Then disaster struck--in a moment of distraction, both forgot to rewind their chronometers and allowed them to run down. Suddenly, they were lost! Based on their last recorded positions, they made a guess and reset their timepieces, but the rest of their journey was clouded by uncertainty. Luckily, they did not have much further to go, and as chance had it, they encountered a British Arctic expedition which took them home. "Fram" broke free from the edge of the ice at about the same time; it is now on public display in Oslo.

Next Stop: #5b.  Coordinates

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Author and curator: David P. Stern
Last updated 21 May 1999