the figure of the earth
the figure of the earth
2026-08-22
Brigadier Martin Hotine is not quite the image of a decorated officer. His name<br>is styled with trailing acronyms that make it no surprise that there is an<br>official portrait, yet in that painting he appears disheveled, his tie far off<br>center in the collar of his jacket. He leans off to one side, not quite like he<br>is sitting for a portrait, but more like he was caught in the middle of<br>something. Photos of the man are often similar: he's distracted, looking down at<br>his desk or staring into space. His mind seems to be elsewhere. Hotine had a lot<br>to think about. His duties in the First and Second World Wars had only been a<br>distraction from the real work of his career: the precise measurement of the<br>whole British Empire.
Late in Hotine's career, he was honored not only by his own country (as a<br>Commander of the Order of the British Empire) but by the United States as well<br>(named an Officer of the Legion of Merit). Most of his awards, though, reflected<br>the technical nature of his work: the Founder's Medal of the Royal Geographical<br>Society, and shortly after his death in 1968, the Gold Medal of the United<br>States Department of Commerce.
Ribbons and medals, though, do not quite capture the breadth of Hotine's work.<br>His greatest memorial is an artifact of his work: squat concrete pillars<br>surmounted by a triangular brass plate. Found atop mountains and hills<br>throughout the United Kingdom, these "trig points," designed by Hotine himself,<br>are the physical references of the Retriangulation of Great Britain. This<br>effort, spanning from 1935 to 1962 with the interruption of WWII, revised an<br>original triangulation (initiated in the 18th century) as the basis for British<br>surveying. Through the course of this effort, Hotine developed methods that<br>would revolutionize the field of geodesy. His collaboration with mapmakers from<br>the United States, a continuation of his wartime surveying for the Allied<br>Forces, set the stage for one of geodesy's most ambitious projects: a<br>measurement taken across the Atlantic Ocean.
Geodesy is the field concerned with the measurement of the Earth. It is perhaps<br>one of the greatest examples of the subtle complexity of the real world:<br>superficially, the measurement of distances and areas is a simple problem. In<br>practice, it is extremely complex, subject to a web of complications that mean<br>that even the most modern efforts should be viewed only as close approximations.
To begin, we have to consider the shape of our planet. This question, "what<br>shape is the Earth?," is a central topic in geodesy known as the "figure of<br>the Earth," and it has occupied mathematicians, cartographers, and astronomers<br>for centuries. Of course we know what shape the Earth is: it is a sphere. Well,<br>that's true to a level of approximation, but one that isn't even close enough<br>for highway construction.
Geodesy's foundations are in the measurement of angles and distances, taken from<br>the Earth's surface—of course, for most of human history, where else would we<br>take them? By measuring the angles between three points and performing some<br>trigonometry, the relative positions of the three points can be determined. This<br>is known as triangulation. The same is true if you measure the distance<br>between three points, known as trilateration, but up until the development of<br>electronics the measurement of very long distances was a far more difficult<br>problem than the measurement of angles.
So, the first triangulation of Great Britain, conducted over some 60 years<br>starting in 1791, measured the angles between mountain and hill peaks. These<br>measurements were taken very precisely using a then-new instrument called a<br>theodolite, which is essentially a telescope coupled to a protractor. By taking<br>enough measurements between enough hilltops, surveyors formed a sort of mesh or<br>web that slowly spread across the country. Eventually, this network of reference<br>points was dense enough that locations of buildings, property claims, and<br>enemy encampments could be stated accurately by their relative position to fixed<br>reference points.
This explanation of geodetic triangulation has omitted a major problem. Three<br>angles or three measurements can precisely define a triangle, but the solution<br>depends on the surface over which the triangle is formed. The math is simplest<br>in a flat plane. Over a sphere, it becomes more complicated but is still well<br>understood. The Earth, though, is not a sphere. It's not even that close.
One of the reasons that it is difficult to define the shape of the Earth is that<br>it is unclear exactly what shape you would refer to. The physical surface of the<br>Earth, its topography, is extremely messy. There are mountains, there are<br>valleys, and the whole thing is the result of long, stochastic processes that<br>left behind something that is not amenable to a mathematical description.<br>Besides, surveyors are often trying to establish where exactly the topography<br>is (e.g. the...