## They all laughed (1): Eratosthenes

They all laughed at Christopher Columbus
When he said the world was round…

(They All Laughed, 1937, lyrics by Ira Gershwin.)

One of those stories that “everybody knows'” is how Christopher Columbus challenged the Bible-based wisdom of his day by insisting that the world was round rather than flat, and to prove it he set out to sail westward from Europe to China. Unfortunately — so the story goes — America got in the way, so instead of proving that the world was round, Columbus became the first European to reach a new continent. Some consolation.

Like most stories that “everybody knows”, this one is largely untrue, and not just because the first Europeans had reached North America some centuries before. More importantly, scholars had accepted for centuries before Columbus that the world was round. Indeed, the first estimates of its size had been made more than 1500 years earlier by the Greek geographer and mathematician Eratosthenes, while remarkably accurate measurements had been made a few centuries before Columbus by the Islamic scholar al-Biruni. Columbus, who had some wildly optimistic geographical theories of his own, was fortunate to run into an unexpected continent: if it hadn’t been there then his ships would have run out of provisions long before they reached China. Sometimes, it seems, planning a research project properly isn’t the best way to get things done.

The work done by Eratosthenes and by al-Biruni is interesting for several reasons. One is that it shows how far one can get with a tough problem using elementary mathematical tools and a lot of patience and ingenuity. Another is that the story links two key periods in the development of mathematics and the natural sciences. In this article, we’ll sketch this story. For simplicity, we’ll describe the mathematics using modern algebraic notation — which will make matters a lot easier for us than it was for either of the protagonists.

Before reading any further, you should spend a few minutes trying to answer the question yourself: given basic measuring equipment and without long-distance travel, how might you go about estimating the size of the Earth? Once you’ve thought that through, read on…

### Ancient Greece: Eratosthenes and the solar method

It is uncertain where and when the theory of a spherical Earth originated, but like many bright ideas it’s often attributed to Pythagoras; certainly it was generally accepted for most of the ancient Greek period. Bearing in mind that they had only explored a small part of the Earth’s surface, the Greeks were convinced as much by metaphysics as by observation. A curved Earth was certainly a useful way to explain why horizons occur, and why the sun rises to different heights in the sky at midday in diffent places, but the fundamental reason for believing in a spherical Earth was that it was mathematically simple. A sphere was a “perfect” shape, so how could the Earth be anything else? Although this isn’t what we think of today as a scientific argument, it provided a sensible basis for asking further questions about the Earth, and it turned out not to be far wrong — modern measurements of the maximum and minimum radii of the Earth only differ by about 0.3%.

C19th reconstruction of Eratosthenes’s map of the world — almost entirely constructed from second-hand reports.

Many people from Pythagoras onward must have asked how large this spherical Earth might be, but one of the first to provide an answer was Eratosthenes of Cyrene (276BC–194BC). Eratosthenes was a scholar with multiple interests: he was unkindly nicknamed “beta” on the basis that he was second-best at everything, and was described snidely as “a mathematician among the geographers, and a geographer among the mathematicians”. Nonetheless, he held the important position of head librarian at the Library of Alexandria, the greatest centre of scholarship in the ancient world.

The crucial piece of information that Eratosthenes possessed came in a letter from a friend in Syene (modern Aswan). He commented that at midday at midsummer, the sun was directly overhead, so its rays reached the bottom of a deep well. This was interesting to a geographer because it meant that Syene was on the tropic — in fact, by modern measurements it is about 30 km away — but Eratosthenes realised he could put it to another use. Syene was believed to be due south of Alexandria, and from other sources Eratosthenes had an estimate of the distance from one to the other: in the units of the day, 5000 stades.

Schematic of Eratosthenes’s method. In the left panel, A indicates Alexandria, S indicates Syene, and O is the centre of the Earth. The Earth’s radius $R = |OA|=|OS|$. The arrows are the (parallel) rays from the sun. The right panel shows the column and shadow used to determine $\theta$.

Alexandria is some way north of the tropic, so at midday at midsummer a vertical column will cast a small shadow to the north. By measuring the length of this shadow and comparing it with the height of the column, one can measure the angle $\theta$ which describes how far the sun is from the vertical. (There are other ways to measure this angle, but this one doesn’t involve looking directly towards the sun!) Assuming that the sun is at a great distance from the Earth, its rays must be roughly parallel, so from a theorem concerning alternate angles, we know that $\theta$ must be the difference in latitude between Syene and Alexandria.

We can now estimate the radius $R$ of the Earth. The arc $AS$ subtended by $\theta$ has a length of 5000 stades, and so we have

$\mbox{circumference} = 2\pi R = \dfrac{2\pi}{\theta}\times 5000\mbox{ stades.}$

Eratosthenes measured the angle $\theta$ as $1/50$ of a full circle ($\approx 7.2^{\circ}$), giving a circumference of 250 000 stades.

This method is nice and simple, and since it relies only on basic geometry and to astronomical observations of the sun it appealed strongly to the ancient Greek mindset. The problem with it is the measurements. The angle $\theta$ has to be large enough to measure it accurately, and in fact Eratosthenes’s value is accurate to within about 2%; but as $\theta$ increases, so does the length of the arc that must be measured. Unless one could rely on highly accurate surveys, the only means available involved counting paces (or the revolutions of a vehicle’s wheels) on the journey and hoping that one’s route was reasonably close to a straight line. Egypt in Eratosthenes’s day was probably the best-surveyed country in the world, thanks to millennia of civilisation and the flooding of the Nile that made it necessary to re-measure agricultural land every year. Nevertheless, it’s reasonable to assume that the measurement of 5000 stades was at best accurate to the single-digit precision with which it’s quoted! The other problem is the assumption that one observation point is due north of the other. Before accurate portable clocks were developed in the eighteenth century, determining longitude was extremely difficult: in fact, Alexandria is about $2.5^{\circ}$ west of Syene, introducing a further minor error.

Sadly, it is hard to evaluate the accuracy of Eratosthenes’s estimate because we don’t know how long a stade is in modern units. (Some modern writers have claimed incredible accuracy for Eratosthenes’s estimate, but this is often because they have defined a stade in terms of the Earth’s known size — a classic piece of, er, circular reasoning.) However, all plausible values for the stade suggest that Eratosthenes was accurate to within 20%: an impressive achievement given the resources available to him.

After the decline of the ancient Greek culture, much of their learning was forgotten in Europe. Much more of it survived in what became the Islamic cultures of North Africa, the Middle East and Central Asia. Thus, in the early ninth century, Caliph al-Mam’un of Baghdad came across a manuscript describing Eratosthenes’s estimate, and became curious about the size of the Earth. (Perhaps, like any powerful monarch, he simply wanted to know what fraction of it he ruled.) His court scholars couldn’t advise him any more accurately than we can today about the length of a stade, so he funded a large-scale project to obtain an updated measurement. This was essentially a systematic application of Eratosthenes’s method, carrying out extensive surveys in the Mesopotamian desert to establish a reliable baseline equivalent to the 5000 stades employed by Eratosthenes. For two centuries, this result remained the definitive estimate and this method was the only game in town. As we’ll see in Part 2, this was about to change…

(Article adapted by DP from an Honours project by Tara Hirshall.)