A little mathematics goes a long, long way; in the adventure recounted below, elementary geometry brought an intrepid band of six men 800 sea miles across the treacherous Southern Ocean, and led to the saving of 28 lives.
Endurance
For eight months, Ernest Shackleton’s expedition ship Endurance was carried along, ice-bound, until it was finally crushed and sank in October 1915. This put an end to the plans of the Irish-born explorer and his companions to cross the Antarctic continent. They salvaged three boats and made their way to Elephant Island, at the tip of the Antarctic Peninsula.
The whaler James Caird approaching South Georgia.
Illustration from Ernest Shackleton’s book, South (William Heinemann, London 1919).
With five companions, Shackleton set out in one of the boats, a whaler called the James Caird, setting course for South Georgia, some 800 nautical miles or about 1500km distant. With unceasing gales, the sea was tempestuous. Navigation depended on sitings taken with a sextant during rare appearances of the Sun. Heavy rollers tossed the boat about, making it difficult to sight the horizon. The process was described by navigator Frank Worsley as “a merry jest of guesswork”.
Schematic diagram of a marine sextant used for navigation.
Image from Wikimedia Commons.
Navigation
Celestial navigation uses angular measurements between celestial bodies, primarily the Sun, and the horizon to determine position. Latitude is measured by “shooting the Sun” with a sextant. The horizon and the lower limb of the Sun are aligned in a split mirror, viewed through a telescope. The altitude of the Sun can then be read from an indicator on the sextant arc.
From the altitude of the Sun, the latitude of the observer can be calculated. The geometry is straightforward: looking at the figure below, we can see that, since the rays from the Sun can be taken as parallel, the corresponding angles β (beta) and γ (gamma) are equal. But β = 90°- α and γ = φ – δ. Therefore, the latitude φ (phi) is given by
φ = 90° + δ – α
where α (alpha) is the Sun’s altitude read from the sextant and δ (delta) is the latitude, or declination, of the Sun. This last depends on the date and time, and is given in the Nautical Almanac.
Angles used to calculate the latitude. Alpha (α) is the altitude of the Sun measured with the sextant, delta (δ ) is the latitude (declination) of the Sun, obtained from the Nautical Almanac and phi (φ) is the latitude of point P.
To get the longitude, a clear shot of the Sun at local noon is required. The navigator tracks the solar altitude to determine the exact time when the Sun reaches its highest point. This is local apparent noon. The chronometer is set to Universal Time (UTC or GMT). As the Earth rotates in 24 hours, the Sun appears to move westward 15 degrees in each hour. Thus, if the chronometer reads 15:00 GMT, local noon is three hours behind Greenwich and the longitude is 45° west.
There are a number of corrections and refinements not described here, and the navigator uses a variety of other techniques. But the most important measurement is the “noon sight”. A modern online converter gives the distance between Elephant Island (61°S, 55°W) and South Georgia (54°S, 37°W) as 1320 km or about 715 nautical miles. The great circle route is shown below. The initial bearing is 60 degrees but, to avoid pack ice, Shackleton headed north, so the journey was closer to 800 nm.
Great circle route from Elephant Island (61°S, 55°W) to South Georgia (54°S, 37°W). Distance 1320 km or about 715 nautical miles
Rescue
After seventeen days, Shackleton and his companions landed on the west coast of South Georgia. The voyage was a marvel of navigation, one of the greatest boat journeys ever accomplished.
But the trouble was not over yet. Shackleton still had to cross the mountainous interior of the island to reach the whaling station at Stromness and arrange a rescue mission to relieve the men left behind on Elephant Island.
Ultimately, the entire party of 28 men reached the safety of Punta Arenas, Chile in September 1916. The survival of Shackleton and all his companions was described as “a triumph of hope and inspired leadership”. It was also a spectacular example of masterful navigation.
There is an exhibition on the last voyage of Endurance and on this most extraordinary maritime rescue at the Ferry Terminal in Dun Laoghaire: shackletonexhibition.com
ThatsMaths 