Around 240 BC, a man standing in Alexandria with a stick worked out the size of the planet. He never left Egypt. He had no telescope, no clock worth the name, and no instrument more sophisticated than a vertical pole and a way of measuring the angle of its shadow. His answer was within a few percent of the figure we use today.
The man was Eratosthenes of Cyrene, chief librarian at Alexandria, and his calculation of the Earth’s circumference is one of those rare moments where you can follow every step of an ancient scientific argument and find nothing to object to. The reasoning is so clean that it is still taught to schoolchildren. What usually gets lost is how much had to be true before the experiment could work at all.
The Librarian Nobody Called First
Eratosthenes was born around 276 BC in Cyrene, in what is now Libya. He studied in Athens and was summoned to Alexandria by Ptolemy III to tutor the royal heir and run the Library, the largest collection of written knowledge in the ancient Mediterranean. It was the best job in the intellectual world.
His contemporaries were not entirely kind about it. They nicknamed him “Beta”, the second letter of the Greek alphabet, on the grounds that he was second best at everything and first at nothing. He wrote on geography, mathematics, poetry, chronology, and astronomy, and the sneer was that a man who does everything masters nothing. History has been more generous. He invented the discipline he named geographia, produced a method for finding prime numbers that still carries his name, and built a chronology of the world going back to the fall of Troy.
The Library mattered here for a practical reason. Eratosthenes had access to records that nobody else could consult: survey reports, travellers’ accounts, and the measured distances between Egyptian cities. The measurement of the Earth was not the work of a lone genius in a field. It was the work of an administrator who knew exactly where to find a number.
The Well at Syene
The experiment rests on an observation that had been circulating for some time. In the town of Syene, modern Aswan, there was a well where at noon on the day of the summer solstice the Sun shone straight down to the water without casting any shadow on the sides. The Sun stood directly overhead, at the zenith. Syene sits almost exactly on the Tropic of Cancer, which is why this happens there and not in Alexandria.
Eratosthenes knew that at the same moment in Alexandria, several hundred kilometres to the north, a vertical pole did cast a shadow. He measured the angle between the pole and the direction of the incoming sunlight and found it to be about one fiftieth of a full circle, which is 7.2 degrees.
Everything else follows from two assumptions. The first is that the Earth is a sphere, which educated Greeks had accepted for well over a century. The second is that the Sun is far enough away that its rays arrive effectively parallel. Grant those two things, and the angle of the shadow in Alexandria is not a fact about the Sun at all. It is the angle subtended at the centre of the Earth between Alexandria and Syene.
If that angle is one fiftieth of a circle, then the distance between the two cities is one fiftieth of the circumference of the planet. Eratosthenes had the distance: 5,000 stadia, a figure that came from the professional pacers who measured routes for the Egyptian administration, or possibly from camel caravan travel times. Multiply by fifty and you get 250,000 stadia. He later adjusted this to 252,000, almost certainly because that number divides neatly by 60 and gave him a round 700 stadia per degree.
How Close Did He Get?
This is where honesty is required, because the popular version of the story tends to oversell it.
The problem is the stadion. It was not a standardised unit, and we do not know for certain which one Eratosthenes used. If he worked with the Egyptian stadion of about 157.5 metres, his result comes out at roughly 39,690 kilometres, which is within one percent of the true polar circumference of 40,008 kilometres. If he used the common Attic stadion of about 185 metres, the answer is about 46,600 kilometres, an overestimate of around 16 percent.
Either way it is a remarkable result, but the first version is the one that gets quoted and it may be flattering him. There were also errors baked into his inputs that partially cancelled each other. Syene is not exactly on the Tropic and not exactly due south of Alexandria, and the 5,000 stadia figure was an approximation. The genius of the experiment does not lie in its accuracy. It lies in the realisation that a shadow in one city and a well in another, taken together, are enough to weigh a planet you have never seen from outside.
Geometry as an Instrument
What Eratosthenes was really doing was using geometry as a measuring device. He had no way of reaching the centre of the Earth, so he substituted an angle he could measure for a distance he could not. That substitution is the engine of almost all of observational astronomy that followed.
The intellectual toolkit came from the same world that produced Euclid, who had worked in Alexandria a generation earlier. The proposition Eratosthenes leaned on, that a line crossing two parallel lines makes equal alternate angles, is Proposition 29 of Book I of the Elements. The idea that a set of abstract axioms about lines and angles could be turned on the physical world and made to yield the size of the Earth is the moment mathematics stops being a game and becomes a science.
The same trick, scaled up, is how we still do it. Measure an angle from two known positions and you get a distance you cannot walk. Hipparchus used it on the Moon. Later astronomers used it on the planets, then on the nearer stars, and eventually on other galaxies. Every rung of the cosmic distance ladder is a variation on a librarian in Egypt looking at a shadow.
What Happened to the Number
Eratosthenes was not the only ancient thinker to attempt this. Posidonius produced a smaller figure a century and a half later, using the star Canopus rather than the Sun. Ptolemy adopted a version of the smaller estimate in his Geography, and because Ptolemy’s authority was enormous, the smaller Earth is the one that travelled through the Middle Ages.
This had consequences. A smaller planet means a shorter westward voyage from Europe to Asia. Columbus worked from figures descended from the low estimates, which is a large part of why he believed the crossing was survivable. Had he trusted Eratosthenes, he might never have sailed. One of the more uncomfortable lessons in the history of science is that the correct measurement was available, in writing, for seventeen centuries, and the wrong one won because it had a more famous advocate.
The number was only properly recovered when European surveyors began measuring arcs of meridian directly in the seventeenth and eighteenth centuries. When France defined the metre as a fraction of the Earth’s quadrant, the expeditions sent out to measure the meridian through Paris were doing Eratosthenes’s experiment with better instruments and a much larger budget.
Reading the Ancient Sky
We do not have Eratosthenes’s own account. His book On the Measurement of the Earth is lost, and what survives comes to us through Cleomedes, writing perhaps three centuries later and simplifying as he went. The round numbers in the story, the tidy fiftieth, the clean 5,000 stadia, are probably Cleomedes tidying up a messier original.
That loss is normal for ancient science and it is part of why the works that did survive matter so much. The geometry Eratosthenes used is preserved in Euclid, and the visual reconstruction of the Elements across fourteen posters shows how those propositions are built one on top of another, including the parallel line theorems that make the shadow argument work.
For the astronomical tradition that grew out of this kind of measurement, our six book pack on the history of astronomy collects the works where the method was refined and eventually turned on the whole solar system, from Copernicus and Kepler to Brahe’s observational programme.
A Stick, a Shadow, and a Willingness to Multiply
The reason the story has survived for more than two thousand years is that it is completely transparent. There is no hidden apparatus, no specialist knowledge, no result that has to be taken on trust. Anyone who understands that parallel lines make equal angles can check the argument, and anyone with two friends in distant cities and a sunny day can repeat it. Schools do exactly that every year.
It also demonstrates something that is easy to forget in an age of instruments: the limiting factor in science is very often not the equipment. Eratosthenes had access to nothing his contemporaries lacked. He had a well that people had been staring into for generations and a distance that was already written down in the archives. What he added was the idea that the two facts belonged together.
That is the part that cannot be bought. A pole, a shadow, a number from a filing cabinet, and the willingness to multiply by fifty gave the ancient world the size of its own planet. It would be another eighteen centuries before anyone did appreciably better.
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