Friday, July 19, 2013

An Empire-ical Theorem

Last fall, I spent seven weeks in Abu Dhabi, teaching a course, eating Marie biscuits, looking for British adapters for my US plug points, staring at the desert, and thinking that the Brits had really been around the block. That thought came back to me a week ago when I was visiting Sydney: "The sun never sets over the British Empire." 

Actually, the phrase was used to refer to Spanish and Portuguese colonization:"el imperio en el que nunca se pone el sol." Of course, in Spanish we may replace "se pone" by "se puso" without fear of error, but what about the Brits? Should I employ the past tense?

It has been argued (not unconvincingly, in my opinion) that the sun never set over the British empire because the British empire was in the east and the sun, as we all know, sets in the west. (This argument bears some superficial similarity to the proof of the assertion that Alexander the Great had an infinite number of limbs; see Theorem 2 here, but I digress.) The British claim the Chagos islands in the Indian Ocean (Mauritius disputes this), a matter to which we return below; might the above proof then not be employed still? It turns out, though, that a more comprehensive though admittedly less elegant argument is available.

To this end, I introduce to you my good friend and former Stanford colleague Peter J. Hammond, now at the University of Warwick. Peter's notes were written on the eve of the handover of Hong Kong to China: the end of Empire as widely acknowledged. But the world had not reckoned with Peter.

Sun Sinking over British Empire

by Peter J. Hammond  

"The sun never sets over the British Empire." Or at least it has not since the Victorian era. But is it about to? On 1st July 1997, when Hong Kong ceases to be a U.K. Dependent Territory and becomes part of the People's Republic of China? In December 1997, when it gets very dark in Britain? Or in June 1998, when the sun is at its lowest over scattered Dependent Territories in the southern hemisphere? Or never in the foreseeable future?

These days, with empires clearly politically incorrect, the "British Empire" presumably consists of the U.K. and its "Dependent Territories" (including Hong Kong until 30th June 1997). These territories abound in the longitudes between the Greenwich Meridian and 90 degrees west, with the U.K. itself, Gibraltar, Bermuda, numerous Caribbean islands, Ascension, St. Helena, Tristan da Cunha, the Falkland Islands, South Georgia --- some of these are even dependencies of some others!) So Hong Kong, at 22 deg. N. and 114 deg. E., certainly helped to fill the gap. (This and other geographical data are taken from the Times Atlas of the World, 6th Comprehensive Edition, 1980.)

However, one should not overlook:

(i) the Pitcairn Islands Group (25 deg. S., 130 deg. W.) in the Eastern Pacific, which was settled by mutineers from HMS Bounty;

(ii) the British Indian Ocean Territory, otherwise known as the Chagos Archipelago, including the Diego Garcia military base at 7 deg. S., 72 deg. E. 

Given the location of these two in the southern hemisphere, the sun is clearly at its lowest over the British Empire (and may even set) around 02:00 hours GMT each 21st June, when it is overhead at 23.45 deg. N. and about 150 deg. E. So, on midwinter's day in the southern hemisphere, does the sun set over Pitcairn before it rises over Diego Garcia? That is the question. (It is interesting that the U.K. itself seems irrelevant, and reassuring that so is the British Antarctic Territory between 20 deg. W. and 80 deg. W.)

Because Pitcairn is some way from the Equator, the question is delicate enough to require reasonably precise estimates of the time at which the sun sets at various latitudes. We use the (northern chauvinist) convention that northern latitudes are positive, southern latitudes are negative. The point where the sun is overhead on the Tropic of Cancer, whose latitude is T = 23.45 deg., is at ( cos T, 0, sin T ) in suitable Cartesian co-ordinates where the origin is the centre of the Earth, the North Pole is at (0, 0, 1), and (1, 0, 0) is the point on the Equator where it is mid-day. The point at latitude L whose longitude differs by H from that of ( cos T, 0, sin T ) is ( cos H cos L, sin H cos L, sin L ) in the same co-ordinate system. So the difference H in longitude between where the sun is overhead and where it is on the horizon (rising or setting) must solve the equation

( cos H cos L, sin H cos L, sin L ) . ( cos T, 0, sin T ) = 0   

This implies that cos H = - tan L tan T. Note that H = 90 deg. when L = 0 (at the Equator), and that there is no solution if |L| > A =  90 deg. - T = 66.55 deg. (beyond the Arctic or Antarctic circles). Furthermore, H > 90 deg. when 0 < L < Abut H < 90 deg. when 0 > L > -A.

Rough calculations suggest that S is about 87 deg. at the latitude (-7 deg.) of Diego Garcia, and that S > 75 deg. even at latitude -30 deg., well south of Pitcairn. So each 21st June the sun rises over Diego Garcia when it is overhead at 157 deg. E. (at roughly 01:32 hrs GMT), whereas it does not set over Pitcairn until it is overhead somewhere west of 155 deg. E. (after about 01:40 hrs GMT).

These calculations were then confirmed and made more precise by accessing the public service "sunrise/sunset computer" over the Internet. This can be found at the website:

Its results allow for the refraction of the sun's rays when it is close to the horizon. They indicate that, on 21st June, the sun rises over Diego Garcia at 01:22 hrs GMT, more than half an hour before it sets over Pitcairn at 01:59 hrs GMT.

Thanks to Diego Garcia (uninhabited except temporarily by various U.K. and U.S. military personnel) and to Pitcairn (population now about 50), the British Empire appears safe from sunsets for the time being. (Both these territories have websites, by the way, though that for Diego Garcia is maintained by the U.S. Navy at But the sun will be getting very low over the British Empire at around 01:40 GMT in late June each year....

Also, it seems that the sun could finally set over the British Empire if the sea level were to rise high enough because of global warming. It turns out that Diego Garcia has a mean elevation of only 4 feet above sea level, and a maximum elevation of only 22 feet. Perhaps the U.S. Navy will erect dikes around their strategically located communication facilities...