Eratosthenes made an estimate of the circumference of the earth over 2000 years ago. Using measurements of the angle of shadows in just 2 places, his calculations produce a surprisingly accurate result.

#### Method

Eratothenes lived in Alexandria, which is in Egypt. He knew that in Syene, during the summer solstice at midday, there was a well which had no shadow at the bottom of it. This meant that the sun was directly above Syene.

He made a measurement of the angle of the shadows in Alexandria on the same day at the same time. This was 7.2 degrees. Using rules of angles with parallel lines (sun’s rays will be parallel), he knew that the angle between Alexandria and Syene from Earth’s centre would also be 7.2 degrees (see below).

As 360/7.2 is 50, all Eratosthenes had to them do was multiply the distance between the two cities by 50 to find the circumference of the Earth. He knew the distance between Syene and Alexandria was 5000 stadia and therefore he calculated the circumference of Earth to be 250,000 stadia.

Unfortunately we do not know which value for one stadia he used, as there was no standard measure. Had he used the Egyptian Stadia (157.5m), his value in kilometres was 39375km. The known value today is 40008km, so the percentage error in his calculation was below 2%!

Had the Greek stadia been used in his calculations, his estimate would have been off by 16%, still fairly impressive given the technology available at the time.

#### Significance

At the time, around 250BC, many geographers were attempting to make accurate maps of the world. As a geographer himself, this was Eratosthenes goal. On his map, he recorded all the distances and details that he knew. To make it more useful as well as accurate, he developed the system of latitude and longitude. However, he didn’t know how wide one degree latitude should be. For this, he required the circumference of the Earth.

Eratosthenes also developed the Sieve of Eratosthenes, a significant advancement in mathematics which was used to determine which numbers were prime. This method is still important in mathematics today.

## References

https://www.windows2universe.org/?page=/citizen_science/myw/w2u_eratosthenes_calc_earth_size.html