Eratosthenes

Chapter 1 · Cyrene to Alexandria

The man they called second-best

c. 276 – c. 245 BC · Cyrene · Athens · Alexandria

Eratosthenes was born around 276 BC at Cyrene, a Greek city on the coast of what is now Libya. He studied at Athens, where the philosophical schools were, and wrote poetry and literary criticism as well as mathematics.

Around 245 BC King Ptolemy III of Egypt summoned him to Alexandria to tutor the royal heir, and then made him chief librarian. The Library of Alexandria was the largest gathering of texts anywhere in the world and the post was among the most desirable in Greek learning.

What is striking about him is the range. He worked on geography, mathematics, astronomy, chronology, grammar and poetry, and he was serious in all of them. This did not impress everybody. Contemporaries nicknamed him Beta, the second letter of the Greek alphabet, meaning that he was second-best in every field and first in none. Another nickname, Pentathlos, made the same point more kindly: a competitor in five events, good at all of them, champion at nothing.

He seems to have taken it well, and the nickname turns out to be a bad judgement. The measurement he is remembered for is exactly the kind of result a person only reaches by working in several fields at once. It needs astronomy, to know what a shadow at noon means; it needs geometry, to turn an angle into a fraction of a circle; and it needs geography, to know how far apart two cities are and to care about the answer. A specialist in any one of the three would not have had the other two.

Why this matters

The most famous measurement of the ancient world came from a man his colleagues thought unfocused — because it needed three subjects at once.

A librarian with too many interests, in the greatest library in the world. What would you ask him?

Ask Eratosthenes

• “Did it bother you to be called second-best at everything?”
• “What was it actually like to run the Library of Alexandria?”
• “Why did a mathematician also write poetry and literary criticism?”
• “How did a scholar from Cyrene end up tutoring a king's son?”
• “Which of your subjects did you care about most?”

Chapter 2 · The Shadow and the Well

How to weigh a world you cannot leave

c. 240 BC · Alexandria · Syene

It was reported to him that at Syene, far up the Nile, a vertical rod casts no shadow at noon on the longest day of the year, and the sun can be seen reflected at the bottom of a deep well. At Alexandria on that same day a vertical rod does cast a shadow, and the shadow implies that the sun is about a fiftieth of a full turn away from directly overhead.

From that difference he got the size of the Earth, and the argument is short enough to hold in your head. If the Earth is a sphere, and if the sun is far enough away that its rays arrive effectively parallel, and if Syene lies directly south of Alexandria, then the arc of the Earth's surface between the two cities is the same fraction of the whole circuit as the shadow angle is of a full turn. A fiftieth.

The distance from Alexandria to Syene was reckoned at five thousand stades. Fifty times five thousand is two hundred and fifty thousand stades, and that is the circumference of the Earth.

Everything in that chain is either an observation, a stated assumption, or arithmetic. Nothing is hidden. He did not walk the distance himself — it came from surveyors, or from the standard reckoning of how long the journey took — and he says so. That honesty about where each ingredient came from is, mathematically, the most valuable part of the whole performance, because it lets anybody who comes after him see exactly where the result could go wrong.

Why this matters

It is the oldest complete piece of quantitative reasoning most people ever meet — an observation, four assumptions, one reported distance, and a number.

A shadow in one city, a well in another, and the size of a planet. What would you ask him?

Ask Eratosthenes

• “Walk me through how the shadow gives you the size of the Earth.”
• “What did the measurement have to assume?”
• “Why does it matter that the sun's rays arrive parallel?”
• “Where did the five thousand stades come from?”
• “Could you have checked the answer against anything?”

Chapter 3 · How Good Was It, Really?

Two wrong inputs that partly cancelled

c. 240 BC and after · Alexandria · Syene

The usual claim is that Eratosthenes got the circumference of the Earth right to within a few per cent. That claim needs handling carefully, and the reasons are more interesting than the number.

First, nobody today knows exactly how long his stade was. Several different stades were in use in the ancient world, and depending on which one is meant his answer ranges from very close indeed to noticeably too big. The precision of the modern comparison is therefore an illusion: we are choosing the stade that makes him look good.

Second, two of his inputs are simply wrong. Syene is not exactly on the tropic, so the rod there does cast a small shadow at midsummer noon. And Syene is not directly south of Alexandria; it sits about three degrees to the east. Both errors push the answer, and they push it in opposite directions, so they partly cancel.

Third, he later quoted the figure as 252,000 stades rather than 250,000. The most likely reason is convenience: 252,000 divides neatly by sixty, which made it easier to work with in a system that reckoned in sixtieths. Adjusting a result for arithmetical tidiness is not fraud, but it does quietly suggest a precision the measurement never had.

None of this diminishes him. The method is sound, the assumptions are visible, and the reasoning survives inspection twenty-two centuries later. That is a far better test of a piece of science than whether the number came out lucky.

Why this matters

A famous right answer, examined properly, turns out to rest on two wrong inputs that happened to offset — which is what makes it worth studying.

A celebrated result, and a closer look at what it rests on. What would you ask him?

Ask Eratosthenes

• “How long is a stade, exactly?”
• “Syene is not really due south of Alexandria. Does that break the method?”
• “Why did you change 250,000 to 252,000?”
• “How accurate did you think your own answer was?”
• “Would you rather have a lucky answer or a sound method?”

Chapter 4 · Lines Across the World

The Geographica, and the invention of a position

c. 235 BC · Alexandria

The measurement was, for Eratosthenes, a means to something larger. He wrote the Geographica in three books, and it is the first attempt to describe the whole known world as one system rather than as a chain of coastlines and traveller's tales.

The innovation is the framework. Earlier descriptions located a place by its neighbours: this town lies four days east of that one, along this coast. Eratosthenes drew lines across the world instead — a principal line running east and west through Rhodes, and another running north and south — and located places by where they fell relative to those lines. A place stopped being described and started having a position.

Once you have that, several things become possible at once. Distances can be compared between places nobody has ever travelled between. The map can be drawn to a consistent scale rather than stretched wherever the reports were fullest. And the size of the Earth, which he had measured, tells you how much world there is left over beyond the part anybody had visited — he suggested that a ship sailing west from Spain might reach India, given enough of the ocean.

His successor Hipparchus attacked the book hard, mainly on the accuracy of individual distances, and he had a point: the reports Eratosthenes was working from were often poor. But the framework survived the criticism, and it is the direct ancestor of every coordinate on every map since.

Why this matters

Giving a place a position rather than a description is the idea behind every map reference, every bearing and every satellite fix.

A world laid out under lines for the first time. What would you ask him?

Ask Eratosthenes

• “Why lay lines across a map instead of describing where places are?”
• “How did you decide where the main line should run?”
• “What could you do with the map that nobody could do before?”
• “Did you really think a ship could sail west from Spain to India?”
• “How did you answer Hipparchus when he attacked your distances?”

Chapter 5 · The Sieve, and the End

A method still taught, and a death still argued about

c. 235 – c. 194 BC · Alexandria

Among his mathematical work is a procedure so simple that it is still the first thing anybody is taught about prime numbers. Write out the whole numbers in order. Strike out every second number after two, then every third after three, then every fifth after five, and so on. What is left has been strained out of the list, and what is left is the primes. It is called the sieve of Eratosthenes and it is still used, in essentially unchanged form, as an efficient way of listing primes up to a limit.

He also compiled a chronology of the world, fixing dates for events from the fall of Troy onwards. That may sound dry, but it was an attempt to do for time what the Geographica did for space: build one framework and put everything into it, so that events in different places could be compared.

He corresponded with Archimedes in Syracuse, who thought well enough of him to address the Method to him — a work in which Archimedes explains, unusually, how he actually discovered his results rather than how he proved them. The Cattle Problem, a fiendish arithmetical puzzle, is also traditionally addressed to Eratosthenes and the mathematicians of Alexandria.

He is said to have gone blind in old age and to have starved himself to death rather than live unable to read, around 194 BC. Like a good deal of what is told about ancient lives, that account comes from much later sources and cannot be checked.

Why this matters

The sieve is still in use, and the chronology shows the same instinct as the map: build one framework and put everything in it.

A method that outlived its author by twenty-two centuries. What would you ask him?

Ask Eratosthenes

• “How does your sieve strain out the prime numbers?”
• “Why try to date the whole past from the fall of Troy?”
• “What was Archimedes like to correspond with?”
• “Why did Archimedes tell you how he really found his results?”
• “Which of your works did you expect to last?”

What Eratosthenes changed

Eratosthenes measured the circumference of the Earth around 240 BC from a shadow angle, one reported distance and a short chain of stated assumptions, and the reasoning is still taught because it can still be checked. His Geographica replaced descriptions of places with positions on a framework of lines, which is the ancestor of every map coordinate. His sieve remains a standard way of listing prime numbers. And his practice of naming what a result depends on, before quoting the result, is a habit science took two thousand years to make universal.

A debate that continues

How accurate his figure really was cannot be settled, because the length of his stade is not known and several were in use. Two of his inputs are certainly wrong — Syene is neither exactly on the tropic nor directly south of Alexandria — and their errors partly cancel, so some of the celebrated closeness is luck. His later revision to 252,000 stades looks like a convenience rather than a measurement.

Keep exploring — ask Eratosthenes

• “How much should a result be trusted when its assumptions are known to be slightly wrong?”
• “Is it dishonest to round a measured figure to a convenient number?”
• “Does working across several fields make somebody worse at each, or better at the problem?”

Related lives

• Archimedes — Give Me a Place to Stand
• Euclid — Who Made Proof The Standard
• Adolphe Quetelet — The Average Man

Related themes

Measurement and estimation · Maps and coordinates · Prime numbers