Measuring the radius of the Earth in 240 BC

The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

Emily Dickinson, ‘The Brain’

Most science teachers find that ‘Space’ is one of the most enduringly fascinating topics for many students: the sense of wonder engendered as our home planet becomes lost in the empty vastness of the Solar System, which then becomes lost in the trackless star-studded immensity of the Milky Way galaxy, is a joy to behold.

But a common question asked by students is: How do we know all this? How do we know the distance to the nearest star to the Sun is 4 light-years? Or how do we know the distance to the Sun? Or the Moon?

I admit, with embarrassment, that I used to answer with a casual and unintentionally-dismissive ‘Oh well, scientists have measured them!’ which (though true) must have sounded more like a confession of faith rather than a sober recounting of empirical fact. Which, to be fair, it probably was; simply because I had not yet made the effort to find out how these measurements were first taken.

The technological resources available to our ancestors would seem primitive and rudimentary to our eyes but, coupled with the deep well of human ingenuity that I like to think is a hallmark of our species, it proved not just ‘world-beating’ but ‘universe-beating’.

I hope you enjoy this whistle stop tour of this little-visited corner of the scientific hinterland, and choose to share some these stories with your students. It is good to know that the brain is indeed ‘wider than the sky’.

I have presented this in a style and format suitable for sharing and discussing with KS3/KS4 students (11-16 year olds).

Mad dogs and Eratosthenes go out in the midday Sun…

To begin at the beginning: the first reliable measurement of the size of the Earth was made in 240 BC and it all began (at least in this re-telling) with the fact that Eratosthenes liked talking to tourists. (‘Err-at-oss-THen-ees’ with the ‘TH’ said as in ‘thermometer’ — never forget that students of all ages often welcome help in learning how to pronounce unfamiliar words)

Alexandria (in present day Egypt) was a thriving city and a tourist magnet. Eratosthenes made a point of speaking to as many visitors as he could. Their stories, taken with a pinch of salt, were an invaluable source of information about the wider world. Eratosthenes was chief librarian of the Library of Alexandria, regarded as one of the Seven Wonders of the World at the time, and considered it his duty to collect, catalogue and classify as much information as he could.

One visitor, present in Alexandria on the longest day of the year (June 21st by our calendar), mentioned something in passing to Eratosthenes that the Librarian found hard to forget: ‘You know,’ said the visitor, ‘at noon on this day, in my home town there are no shadows.’

How could that be? pondered Eratosthenes. There was only one explanation: the Sun was directly overhead at noon on that day in Syene (the tourist’s home town, now known as Aswan).

The same was not true of Alexandria. At noon, there was a small but noticeable shadow. Eratosthenes measured the angle of the shadow at midday on the longest day. It was seven degrees.

No shadows at Syene, but a 7 degree shadow at Alexandria at the exact same time. Again, there was only one explanation: Alexandria was ’tilted’ by 7 degrees with respect to Syene.

Seven degrees of separation

The sphericity of the Earth had been recognised by astronomers from c. 500 BC so this difference was no surprise to Eratosthenes, but what he realised that since he was comparing the length of shadows at two places on the Earth’s surface at the same time then the 7o wasn’t just the angle of the shadow: 7o was the angle subtended at the centre of the Earth by radial lines drawn from both locations.

Eratosthenes paid a person to pace out the distance between Alexandria and Syene. (This was not such an odd request as it sounds to our ears: in the ancient world there were professionals called bematists who were trained to measure distances by counting their steps.)

It took the bematist nearly a month to walk that distance and it turned out to be 5000 stadia or 780 km by our measurements.

Eratosthenes then used a simple ratio method to calculate the circumference of the Earth, C:

Then:

The modern value for the radius of the Earth is 6371 km.

Ifs and buts…

There is still some debate as to the actual length of one Greek stadium but Eratosthenes’ measurement is generally agreed to within 1-2% of the modern value.

Sadly, none of the copies of the book where Eratosthenes explained his method called On the measure of the earth have survived from antiquity so the version presented here is a simplified one outlined by Cleomedes in a later book. For further details, readers are directed to the excellent Wikipedia article on Eratosthenes.

Astronomer Carl Sagan also memorably explained this method in his 1980 TV documentary series Cosmos.

You might want to read…

This is part of a series exploring how humans ‘measured the size of the sky’:

Part 2: How Aristarchus measured the distance between the Earth and the Moon

Part 3: How Aristarchus measured the distance between the Earth and the Sun

9 thoughts on “Measuring the radius of the Earth in 240 BC

  1. paulmartin42 May 23, 2021 / 5:03 pm

    “Astronomer Carl Sagan also memorably explained this method in his 1980 TV documentary series Cosmos.”
    Never saw the original and with the poor (relative) quality I skipped this. To be fair am not a fan of Brian Cox (both of them). My feeling is that the “colour” padding detracts from the explanation, and after a while “Bootiful” grates.

    Bronowski’s Ascent of Man: S=k.log.W and the tribute at the end in the pool of water at Auchwitz to Leo Szilard show how such should be done.

  2. e=mc2andallthat May 23, 2021 / 8:23 pm

    Sagan’s Cosmos is well worth watching! I have only vague memories of Bronowski’s Ascent of Man but I think I may track it down and try it again….

    • paulmartin42 May 24, 2021 / 5:23 am

      JACOB BRONOWSKI – The Tragedy of Mankind 8 mins on Youtube

      And his appearance on Parkinson was good

  3. Hallouminatus May 27, 2021 / 2:40 pm

    “the longest day of the year (March 21st by our calendar)”
    I think you mean June,
    “since he was comparing the length of shadows at two places on the Earth’s surface at the same time then the 7degrees wasn’t just the angle of the shadow: 7degrees was the angle subtended at the centre of the Earth by radial lines drawn from both locations.”
    Only true if one place is due South of the other.

    • e=mc2andallthat May 27, 2021 / 7:44 pm

      Oops! How did I miss the longest day? Thanks for pointing it out 🙂 The blog outlines the simplified version of Eratosthenes’ method presented by Cleomedes. Alexandia and Syene are not on the same meridian but are quite close which means the answer is qutie close to the modern value (more by luck than judgement, possibly)

    • e=mc2andallthat May 28, 2021 / 6:05 pm

      Just one further thought: since they are comparing the height of the Sun at local noon in their respective locations (but not at the same time) then the 7 degrees is effectively the angular separation there would be if they were on the same meridian…at least, I think that’s the case. The method will work as long as the measured distance between them used in the calculation is not dissimilar to their north-south displacement.

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