Did Eratosthenes really measure the size of the earth?

Last Thursday was Summer Solstice in the Northern Hemisphere and The Guardian chose to mark the occasion with an article by astrophysicist turned journalist and novelist, Stuart Clark, who chose to regale his readers with a bit of history of science. The big question was would he get it right? He has form for not doing so and in fact, he succeeded in living up to that form. His article entitled Summer solstice: the perfect day to bask in a dazzling scientific feat, recounted the well know history of geodesy tale of how Eratosthenes used the summer solstice to determine the size of the earth.

Eratosthenes of Cyrene was the chief librarian at the great library of Alexandria in the third century BC. So the story goes, he read in one of the library’s many manuscripts an account of the sun being directly overhead on the summer solstice as seen from Syene (now Aswan, Egypt). This was known because the shadows disappeared at noon, when the sun was directly overhead. This sparked his curiosity and he set out to make the same observation in Alexandria. On the next solstice, he watched as the shadows grew small – but did not disappear, even at noon.

The length of the shadows in Alexandria indicated that the sun was seven degrees away from being directly overhead. Eratosthenes realised that the only way for the shadow to disappear at Syene but not at Alexandria was if the Earth’s surface was curved. Since a full circle contains 360 degrees, it meant that Syene and Alexandria were roughly one fiftieth of the Earth’s circumference away from each other.

Knowing that Syene is roughly 5000 stadia away from Alexandria, Eratosthenes calculated that the circumference of the Earth was about 250,000 stadia. In modern distance measurements, that’s about 44,000km – which is remarkably close to today’s measurement of 40,075km.

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Eratosthenes also calculated that the tilt of the Earth’s polar axis (23.5 degrees) is why we have the solstice in the first place.

Illustration showing a portion of the globe showing a part of the African continent. The sunbeams shown as two rays hitting the ground at Syene and Alexandria. Angle of sunbeam and the gnomons (vertical pole) is shown at Alexandria, which allowed Eratosthenes’ estimates of radius and circumference of Earth.
Source: Wikimedia Commons

Whilst it is correct that Eratosthenes was chief librarian of the Alexandrian library one should be aware that the Mouseion (Shrine of the Muses, the origin of the modern word, museum), which housed the library was more akin to a modern academic research institute than what one envisages under the word library. Eratosthenes was according to the legends a polymath, astronomer, cartographer, geographer, mathematician, poet and music theorist.

From the information that during the summer solstice the sun was directly overhead in Syene at noon, and cast no shadows and that a gnomon in Alexandria 5000 stadia north of Syene did cast a shadow, Eratosthenes did not, and I repeat did not, realise that the Earth’s surface was curved. Eratosthenes knew that the Earth’s surface was curved, as did every educated Greek scholar in the third century BCE. Sometimes I get tired of repeating this but the first to realise that the Earth was a sphere were the Pythagoreans in the sixth century BCE. Aristotle had summarised the empirical evidence that showed that the Earth is a sphere in the fourth century BCE, in writings that Eratosthenes, as chief librarian in Alexandria, would have been well acquainted with. Put simply, Eratosthenes knew that he could, using trigonometry, calculate the diameter of the Earth’s sphere with the data he had accumulated, because he already knew that it was a sphere.

The next problem with the account given here is one that almost always turns up in popular version of the Eratosthenes story; there wasn’t just one measure of length in the ancient Greek world known as a stadium but quite a collection of different ones, all differing in length, and we have absolutely no idea which one is meant here. It is in the end not so important as all of them give a final figure with 17% or less error compared to the true value, which is for the method used quite a reasonable ball park figure for the size of the Earth. However this point is one that should be mentioned when recounting the Eratosthenes story. Eratosthenes may or may not have calculated the tilt of the Earth’s axis but this is of no real historical significance, as the obliquity of the ecliptic, as it is also known, was, like the spherical shape of the Earth, known well before his times.

An astute reader might have noticed that above I used the expression, according to the legends, when describing Eratosthenes’ supposed talents. The problem is that everything we know about Eratosthenes is hearsay. None of his alleged many writings have survived. We only have second hand reports of his supposed achievements, most of them centuries after he lived. This raises the question, how reliable are these reports? A comparable situation is the so-called theorem of Pythagoras, well known to other cultures well before Pythagoras lived and only attributed to him long after he had died.

The most extreme stance is elucidated by historian of astronomy, John North, in his one volume history of astronomy, Cosmos:

None of Eratosthenes’ writings survive, however, and some have questioned whether he ever found either the circumference of the Earth, or – as is often stated – the obliquity of the ecliptic, on the basis of measurements.

So what is our source for this story? The only account of Eratosthenes’ measurement comes from the book On the Circular Motions of the Celestial Bodies by the Greek astronomer Cleomedes and with that the next problems start. It is not actually known when Cleomodes lived. On the basis of his writings Thomas Heath, the historian of Greek mathematics, thought that text was written in the middle of the first century BCE. However, Otto Neugebauer, historian of ancient science, thought that On the Circular Motions of the Celestial Bodies was written around 370 CE. Amongst historians of science the debate rumbles on. North favours the Neugebauer date, placing the account six centuries after Eratosthenes’ death. What exactly did Cleomodes say?

The method of Eratosthenes depends on a geometrical argument and gives the impression of being slightly more difficult to follow. But his statement will be made clear if we premise the following. Let us suppose, in this case too, first, that Syene and Alexandria he under the same meridian circle, secondly, that the distance between the two cities is 5,000 stades; 1 and thirdly, that the rays sent down from different parts of the sun on different parts of the earth are parallel; for this is the hypothesis on which geometers proceed Fourthly, let us assume that, as proved by the geometers, straight lines falling on parallel straight lines make the alternate angles equal, and fifthly, that the arcs standing on (i e., subtended by) equal angles are similar, that is, have the same proportion and the same ratio to their proper circles—this, too, being a fact proved by the geometers. Whenever, therefore, arcs of circles stand on equal angles, if any one of these is (say) one-tenth of its proper circle, all the other arcs will be tenth parts of their proper circles.

Any one who has grasped these facts will have no difficulty in understanding the method of Eratosthenes, which is this Syene and Alexandria lie, he says, under the same mendian circle. Since meridian circles are great circles in the universe, the circles of the earth which lie under them are necessarily also great circles. Thus, of whatever size this method shows the circle on the earth passing through Syene and Alexandria to be, this will be the size of the great circle of the earth Now Eratosthenes asserts, and it is the fact, that Syene lies under the summer tropic. Whenever, therefore, the sun, beingin the Crab at the summer solstice, is exactly in the middle of the heaven, the gnomons (pointers) of sundials necessarily throw no shadows, the position of the sun above them being exactly vertical; and it is said that this is true throughout a space three hundred stades in diameter. But in Alexandria, at the same hour, the pointers of sundials throw shadows, because Alexandria lies further to the north than Syene. The two cities lying under the same meridian great circle, if we draw an arc from the extremity of the shadow to the base of the pointer of the sundial in Alexandria, the arc will be a segment of a great circle in the (hemispherical) bowl of the sundial, since the bowl of the sundial lies under the great circle (of the meridian). If now we conceive straight lines produced from each of the pointers through the earth, they will meet at the centre of the earth. Since then the sundial at Syene is vertically under the sun, if we conceive a straight line coming from the sun to the top of the pointer of the sundial, the line reaching from the sun to the centre of the earth will be one straight line. If now we conceive another straight line drawn upwards from the extremity of the shadow of the pointer of the sundial in Alexandria, through the top of the pointer to the sun, this straight line and the aforesaid straight line will be parallel, since they are straight lines coming through from different parts of the sun to different parts of the earth. On these straight lines, therefore, which are parallel, there falls the straight line drawn from the centre of the earth to the pointer at Alexandria, so that the alternate angles which it makes arc equal. One of these angles is that formed at the centre of the earth, at the intersection of the straight lines which were drawn from the sundials to the centre of the earth; the other is at the point of intersection of the top of the pointer at Alexandria and the straight line drawn from the extremity of its shadow to the sun through the point (the top) where it meets the pointer. Now on this latter angle stands the arc carried round from the extremity of the shadow of the pointer to its base, while on the angle at the centre of the earth stands the arc reaching from Syene to Alexandria. But the arcs are similar, since they stand on equal angles. Whatever ratio, therefore, the arc in the bowl of the sundial has to its proper circle, the arc reaching from Syene to Alexandria has that ratio to its proper circle. But the arc in the bowl is found to be one-fiftieth of its proper circle.’ Therefore the distance from Syene to Alexandria must necessarily be one-fiftieth part of the great circle of the earth. And the said distance is 5,000 stades; therefore the complete great circle measures 250,000 stades. Such is Eratosthenes’ method. (This is Thomas Heath’s translation) 

You will note that Cleomedes makes no mention of Eratosthenes determining the spherical shape of the Earth through his observations but writes very clearly of great circles on the globe, an assumption of spherical form. So where does Stuart Clark get this part of his story? In his article he tells us his source:

I first heard the story when it was told by Carl Sagan in his masterpiece TV series, Cosmos.

The article has a video of the relevant section of Sagan’s Cosmos and he does indeed devote a large part of his version of the story to explaining how Eratosthenes used his observations to determine that the Earth is curved. In other words Stuart Clark is just repeating verbatim a story, which Carl Sagan, and or his scriptwriters, made up in 1980 without taken the trouble to verify the accuracies or even the truth of what he saw more than thirty years ago. Carl Sagan said it, so it must be true. I have got into trouble on numerous occasions by pointing out to Carl Sagan acolytes that whatever his talents as a science communicator/populariser, his history of science was to put it mildly totally crap. Every week he pumped his souped-up versions of crappy history of science myths into millions of homes throughout the world. In one sense it is only right that Neil deGasse Tyson presented the modern remake of Cosmos, as he does exactly the same.