At noon on the summer solstice, in the Egyptian frontier town of Syene — modern Aswan — something odd happened to the light. The sun stood so exactly overhead that vertical columns cast no shadow, and, as travellers reported, its reflection could be seen shining straight up from the water at the bottom of a deep well. Nearly 800 kilometres to the north in Alexandria, at the same noon on the same day, shadows did not vanish. A vertical rod — a gnomon — cast a short but definite shadow. To almost anyone, an idle curiosity. To the chief librarian of the greatest library on Earth, it was a ruler laid across the planet.
Eratosthenes of Cyrene, born around 276 BC in what is now Libya, was the third head of the Library of Alexandria, appointed by Ptolemy III to tutor his heir and mind the world's knowledge. He was a polymath of almost comic range — mathematician, poet, geographer, astronomer, athlete of scholarship — and his colleagues teased him for it with the nickname "Beta": second-best at everything, first at nothing. Others, more kindly, called him "Pentathlos", the all-rounder. The nickname has not aged well, because the measurement he made in around 240 BC is among the most elegant experiments in the whole history of science.
One-fiftieth of a circle
The reasoning takes three steps, each simple, together devastating. First: the sun is so far away that its rays arrive at Syene and Alexandria effectively parallel. Second: if the same parallel rays strike Syene vertically but hit Alexandria at an angle, that angle can only come from the curvature of the Earth between the two cities. Eratosthenes measured the shadow angle at Alexandria as about 7.2 degrees — one-fiftieth of a full circle. Third: if the arc between Alexandria and Syene subtends one-fiftieth of the Earth's circumference, then the whole circumference is simply fifty times the distance between them. That distance was reckoned at 5,000 stadia — by tradition measured by bematists, professional surveyors trained to walk in strictly equal paces. Multiply by fifty: the Earth is 250,000 stadia around. Ancient sources report that he later adjusted the figure to 252,000, conveniently divisible into 700 stadia per degree.
How good was the answer? Here honesty demands a shrug, because nobody is certain how long his stadion was. If he used the Egyptian stade of about 157.5 metres, his circumference comes out near 39,400 kilometres — within about 2 per cent of the true meridional figure of 40,008 kilometres. If he used the longer Attic stade of about 185 metres, the error grows to roughly 16 per cent — still astonishing for a man with a stick. Scholars also note that our main account comes from Cleomedes, writing a simplified version centuries later, and that several of Eratosthenes' assumptions were conveniently wrong in cancelling directions: Syene is not quite on the Tropic of Cancer, it is not quite due south of Alexandria, and 5,000 stadia was a suspiciously round number. The method, though, was flawless.
The librarian's other lives
Measuring planets was almost a sideline. Eratosthenes gave mathematics the sieve that still bears his name — the ancient algorithm for finding prime numbers by systematically crossing out multiples. He effectively founded a discipline and named it, coining the word "geography", and drew a map of the known world gridded with parallels and meridians. He measured the tilt of the Earth's axis with respectable accuracy and proposed adding a leap day to the calendar every fourth year. In an age when knowledge lay scattered across scrolls in a dozen languages, he was paid, in effect, to know everything — and came closer than most humans ever have.
His number had a strange afterlife. Later Greek thinkers, notably Posidonius, produced smaller estimates of the Earth, and the astronomer Ptolemy adopted a figure far below Eratosthenes' in his hugely influential Geography. Centuries on, Christopher Columbus leaned on the small-Earth tradition, convincing himself that Asia lay a manageable sail west of Spain. Had he trusted the librarian of Alexandria instead, the true distance might have kept him in port — and the Americas would have been reached on some other calendar. Rarely has a dispute about units mattered more to world history.
Eratosthenes died in Alexandria around 194 BC; ancient tradition, impossible to verify, says that having gone blind and unable to read, he starved himself to death. He had never needed to leave Egypt to take the planet's measure. A stick, a well, one-fiftieth of a circle — and the nerve to trust a sunbeam all the way around the world.
Quiz nuggets
- At noon on the summer solstice the sun lit the bottom of a well at Syene (modern Aswan), while Alexandria's shadows showed an angle of 7.2 degrees — one-fiftieth of a circle.
- Eratosthenes multiplied the 5,000-stadia Alexandria–Syene distance by 50 to reach 250,000 stadia, around 240 BC.
- If he used the Egyptian stade of about 157.5 metres, his result was within about 2 per cent of the true 40,008 km circumference.
- Eratosthenes was the third chief librarian of Alexandria, coined the word "geography", and devised the prime-number sieve named after him.
- Colleagues nicknamed him "Beta" — supposedly second-best at everything.