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Use Math for Ocean Navigation!

The role of math over centuries of sailing the seas

Beneath the August sun, lakes and saltwater bays fill with recreational sailboats. Even summer sailors rely on basic geometry to “tack” toward a destination by changing their sail angles in relation to the wind. And this just skims the surface of how math has intersected with sailing since antiquity.

Ocean-going vessels today navigate by GPS, which is global positioning by use of satellites. The concept arose in 1973 and was first applied to navigation in 1978.

But how did sailors find their way before GPS? Let’s take a look at nautical tools and techniques devised by math ingenuity starting far back in time.

The first GPS satellite designs were based on an idea called triangulation. This has ancient origins. In the 6th century B.C., the Greek philosopher Pythagoras discovered a math theorem that became the basis of trigonometry––which enabled substantial advancement in ship navigation. Sailors have long used triangulation to find their location at sea by taking measurements from three remote points, and determining where they are within that triangle.

This navigation instrument invented in China around 100 B.C. enabled ships to sail over the horizon without risk of being lost at sea. It has a “travel arrow” that points in the direction you intend to go. A floating metal needle always aligns to the magnetic north pole of the Earth. This type of compass finally found its way to the Mediterranean region of Europe in the late 1200s.

Polynesians sailing as early as 1200 B.C. were the first to navigate the open ocean––and the first to use celestial navigation. These sailors developed a precise scientific system related to the positions of stars. They navigated as well by observation of ocean currents and swells, wind and cloud patterns, the rise and fall of the sun, daylight colors of the sea and sky, and flight patterns of birds.

Ancient Polynesians also created a star compass by subdividing the sky into corresponding quadrants. A star rises at a specific point on the horizon. It will then set in the opposite quadrant from where it arose. The early seafarers of the South Pacific were able to measure stellar elevations. They used these measurements to determine their position in relation to distant specific islands.

The Vikings were medieval master shipbuilders who sailed across the North Atlantic between 900 and 1200 A.D. to oversee their colonies in Iceland and Greenland. They used a highly accurate sundial-like instrument called a sun compass to guide their voyages.

When Christopher Columbus sailed from Spain in 1492 he used this process of calculating a ship’s position at sea. Dead reckoning navigation is based on time, direction, distance traveled, and velocity––rather than on celestial bodies.

To pinpoint an ocean location, sailors need to know both its latitude and longitude. Latitudes are parallel lines that circle the Earth east/west. Longitude meridians run north/south. Together, they form an invisible grid over the Earth that navigators use as a nautical map.

In 1730, a British mathematician named Jesse Ramsden invented the sextant, a handheld telescopic instrument that changed the face of European celestial navigation. Mariners used this device to measure the angles of the sun, moon, and horizon. They could then determine latitude. But calculating longitude was more difficult. That required being able to accurately tell time at sea.

In the 17th century, a Dutch mathematician named Christiaan Huygens tried to make a clock navigators could use to find longitude at sea. Toward this end, he invented, amazingly, the first pendulum clock in 1656. Pendulum clocks could indeed keep accurate time on land––but not on a heaving ship’s deck. It would take another hundred years for great math thinkers to finally solve this problem.

In 1761, an English watchmaker named John Harrison invented the marine chronometer. This precision clock kept accurate time on a boat, and sailors could at last determine longitude. They could do this by calculating the difference between the time on their ship and the time at a fixed land position. Navigators today apply spherical trigonometry to measure time and determine longitude.

Enjoy this short animation video that summarizes the history of sailing navigation:

And here’s one that illustrates logarithms, a calculation method invented in the 1600s that figured powerfully in explorations on the high seas.