# Eratosthenes’ calculation of Earth’s circumference

An illustration showing how Eratosthenes calculated the circumference of the Earth without leaving Egypt. He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He measured the Sun’s angle of elevation at noon on the same day in Alexandria. The method of measurement was to make a scale drawing of that triangle which included a right angle between a vertical rod and its shadow. This turned out to be 1/50th of a circle. Taking the Earth as spherical, and knowing both the distance and direction of Syene, he concluded that the Earth’s circumference was fifty times that distance.

His knowledge of the size of Egypt was founded on the work of many generations of surveying trips. Pharaonic bookkeepers gave a distance between Syene and Alexandria of 5,000 stadia (a figure that was checked yearly). Some say that the distance was corroborated by inquiring about the time that it took to travel from Syene to Alexandria by camel. Carl Sagan says that Eratosthenes paid a man to walk and measure the distance. Some claim Eratosthenes used the Olympic stade of 176.4 m, which would imply a circumference of 44,100 km, an error of 10%, but the 184.8 m Italian stade became (300 years later) the most commonly accepted value for the length of the stade, which implies a circumference of 46,100 km, an error of 15%. It was unlikely, even accounting for his extremely primitive measuring tools, that Eratosthenes could have calculated an accurate measurement for the circumference of the Earth. He made two important assumptions (neither of which is perfectly accurate): 1- That the distance between Alexandria and Syene was 5000 stades, and 2- That the Earth was a perfect sphere.

Eratosthenes later rounded the result to a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia, likely for reasons of calculation simplicity as the larger number is evenly divisible by 60. In 2012, someone repeats Eratosthenes’ calculation with more accurate data, the result is 40,074 km, which is 66 km different (0.16%) from the currently accepted polar circumference of the Earth.

Image: wikipedia