Where in the world? The mathematics of navigation

By: Dr David Rooney, Keeper of Technologies and Engineering
Reading time: Approx 10 minutes

In the eighteenth century the lives of thousands of sailors depended on safe navigation at sea. How could they find out where they were? The answer lay in the stars – and in mathematics.

The longitude problem

There was an explosion of trade in the sixteenth and seventeenth centuries between Britain and America, China, South Asia and South-East Asia. Vast trade empires built up as new trade routes allowed new products to flow between Europe and the rest of the world.

But there was a problem. Everything was transported by sea, and millions of tonnes of cargo as well as thousands of sailors’ lives were lost each year in shipwrecks caused by navigational errors.

Until the 1760s there was no reliable way to navigate out of sight of land. Latitude (the ship’s north–south position) was relatively easy to find from the Sun. The problem was finding the longitude (the ship’s east–west position). 

The nation that ruled the waves could rule the world. So when solutions to the longitude problem were developed in the eighteenth century, new opportunities for trade and travel opened up across the globe.

Nothing is so much wanted and desired at sea as the discovery of the longitude ... In the judgment of able mathematicians and navigators, several methods have already been discovered, true in theory, though very difficult in practice
— Longitude Act, 1714

Using time to navigate space

Longitude navigation was all about time. As the Earth rotates on its axis once per day, the time differences between different places around the world could be used to work out the angular distance, or longitude, between them.

It was easy to find local time on board a ship using the Sun. The difficulty was knowing the time at a fixed reference point on land, such as Greenwich in London, at the same instant. 

are accurate portable timekeepers designed to work on ships. The first marine timekeepers were developed by John Harrison in the 1750s and 1760s. The term ‘chronometer’ was first applied to the devices in 1780 by John Arnold.

The first practical marine timekeeper for longitude navigation was completed by John Harrison in 1759

Several techniques were developed in the eighteenth century to address this problem. Carrying an accurate timekeeper, known as a chronometer, on board was the most widely used. However, chronometers were expensive and fragile. If they stopped, Greenwich time was lost.

There was an alternative that used simpler technology. The lunar distance technique was cheap, accurate and reliable, and was widely used from the 1760s onwards. Unlike the chronometer method, the lunar distance technique found Greenwich time from the stars, reducing the chance of errors. However it involved complicated, time-consuming mathematics.

Taking a lunar distance

Navigators measured the angular distance between the Moon and the Sun or stars in a navigational technique known as ‘taking a lunar distance’  

It was a cool October morning in the early 1770s, and the navigating officer of an East India Company ship 500 miles off Africa’s north-west coast was about to work out the ship’s exact location.

Every clear night was the same. With a fine brass sextant in hand, the officer measured the angles between the Moon and the Sun or certain stars, noting down the time and repeating his measurements to ensure accuracy.

These measurements were known as lunar distances. They were hard enough to find as the ship pitched and rolled, but to get a fix on the ship's position also required hours of mathematical calculation.

The mathematics of lunar distances

After navigators had measured the angles between the moon and the Sun or certain stars, they then performed lengthy calculations. This took into account atmospheric conditions and other optical distortions that would affect the accuracy of their measurements.

Sextants were hand-held astronomical instruments for measuring angles accurately

Navigators could look up Greenwich time in special tables published each year

Finally, the navigators turned to nautical almanacs. These books were published annually and contained tables of numbers calculated by astronomers and other experts. The tables allowed Greenwich time to be looked up for any set of sextant readings.

The numbers contained in nautical almanacs were the result of extensive mathematical labour carried out at observatories and other mathematical institutions. Much of the laborious work was done by women and children, known as computers, working from their homes. 

The end of the longitude problem?

Was the longitude problem solved in the 1760s? Both the lunar distance technique and the chronometer method were widely used to find longtitude right up to the late-twentieth century. They saved countless lives and enabled empires to expand.

Today, however, the advent of satellite navigation such as GPS means that sailors have largely lost their hard-won skills in navigating using time, mathematics and the Moon and stars. But what if the electronics go wrong and the signals are jammed?

With piracy and terrorism ever-growing threats in the world’s seaways, the vulnerabilities of satellite navigation are prompting some sailors to learn celestial navigation techniques as backup once more.

Header image: The skeletal figure of Death sits on a beach with two shipwrecked sailors: Handcoloured copperplate drawn and engraved by Thomas Rowlandson from The English Dance of Death, Ackermann, London, 1816 © Florilegius / Science & Society Picture Library 

Rotational photography: produced by the Science Museum Group Digital Lab in partnership with Samsung.