One of the fascinating aspects of my voyage to Antarctica was learning about the various explorers from the Heroic Age and their expeditions, and then seeing some of the huts that they had constructed for these expeditions. We had an excellent historian on board, and we learned about James Clark Ross, Carsten Borchgrevink, Robert Falcon Scott, Ernest Shackleton, and Roald Amundsen, all of whom led expeditions and all of whom set and held records for having traveled the furthest south on the planet:

• On January 23, 1842, Ross and his team reached 78°09'30"S.
• On February 16, 1900, Borchgrevink reached 78°50'S.
• On December 30, 1902, Scott reached 82°11'S.
• On January 9, 1909, Shackleton reached 88°23'S (only 112 miles from the south pole).
• On December 14th, 1911, Amundsen became the first person to reach the south pole, located at 90°S.

It was fascinating to learn about these men (and sadly, there was a long history of women not being allowed on expeditions and even working in research stations in Antarctica, which has only improved since the 1990s). But hearing about these expeditions also made me want to ask a lot of questions. And one of the biggest questions I had was the following: how were these explorers about to determine how far south they had traveled without GPS?

It turns out that there were several ways to determine one's longitude using the tools that were available at that time, and one of these methods involved tracking the position of the sun. It turns out that the axis upon which the earth rotates is actually tilted at a 23°26.3' angle with respect to its orbit around the sun.

While it's true that the direction of the tilt changes relative to the sun as the earth rotates around the sun, astronomers have been able to calculate the effective angle of tilt is down to the hour. This is called the declination angle. This information can be found in a nautical almanac, and it is essential in using the sun to calculate one's longitude. The method of calculating one's latitude involves the following steps:

1. Use a sextant to measure the angle that the sun makes above the horizon at its highest point. We'll call this the altitude and represent it as A. This usually occurs near noon local time, and so navigators usually took several measurements at five-minute intervals to determine the highest.
2. Look up the declination angle D in a nautical almanac.
3. The observer's latitude L can be calculated using the declination angle and the sextant angle. There are three cases:

• The observer is in the opposite hemisphere as the declination: L = 90° - A - D
• The observer is in the same hemisphere as the declination and they are further from the equator: L = 90° - A + D
• The observer is in the same hemisphere as the declination and they are closer to the equator: L = D + A - 90°

So as an example, suppose one of the Antarctica explorers from the Heroic Age had reached their furthest south point. They used their sextant to determine that the altitude of the sun was 23°35' at its highest, and their almanac told them that the solar declination was 12°25'S. What was their latitude? And which explorer was it?

Since the solar declination was 12°25'S but the observer was further south (as Antarctic territory has latitudes of at least 60°S), we use the second case above. We have L = 90° - A + D = 90° - 23°35' + 12°25' = 78°50'S. This corresponds with Borchgrevink's furthest south record.

One quick note: when celestial navigation is done in the real world, corrections are usually made for the height above sea level of the eye of the observer and refraction of light in the atmosphere. These have been left out to simplify our calculations.

Sample Problems

1. Suppose you're sailing in the Ross Sea. Your sextant shows that the sun's highest angle is 34°25' from the horizon, and your nautical almanac shows that the solar declination is 16°50'S. What is your latitude?

2. Suppose you're sailing off the east coast of Greenland. Your sextant shows that the sun's highest angle is 15°40' from the horizon, and your nautical almanac shows that the solar declination is 2°10'S. What is your latitude?

3. Suppose you're adrift off the coast of the Philippines, somewhere between 6°N and 18°N. Your sextant shows that the sun's highest angle is 81°40' from the horizon, and your nautical almanac shows that the solar declination is 22°15'N. What is your latitude?

4. If you can get access to a sextant or a sextant app, use it to calculate the angle when the sun is at its highest from your home. Look up the solar declination from a nautical almanac and use this to calculate your latitude. Then check how accurate you were online.