Egyptians used mathematics for practical purposes: surveying, trading, construction. The Greeks were the first to do mathematics to do mathematics, without practical and immediate application in everyday life. Eratosthenes (3rd century BC) was one of them.
How did Eratosthenes calculate the circumference and radius of the earth ?
Eratosthenes supposed that the earth was round! Remember that it will be necessary to wait for the rebirth in Europe to accept the rotundity of our planet. This intuition came to him by observing the shadow of the earth on the moon during an eclipse. He observed to Syene that at noon the rays of the sun succeeded in illuminating the bottom of a well. The sun was therefore exactly vertical (Zenith).
Second great intuition: he supposed that the sun was far enough for his rays to arrive parallel to all points of the earth and, in particular, to Alexandria where he had the same experience. At noon, this time, the rays did not light up the bottom of the well. They formed with him an angle of 7 °.
Eratosthenes knew the distance between Syene and Alexandria: 5000 stadia (a stage of 160 m). The arc between the two cities was 7 °. Thus 1 ° of circumference gave 5000/7 = 714 stades and 360 ° of circumference gave 5000/7 x 360 = 250 000 stades, that is to say 40 000 km. The radius of the earth was therefore R = 40 000 / 2π = 6400 km.
We now know that the average value of the radius of the earth (the earth is not round but oval) is R = 6 371,008 km.
The precision of Eratosthenes was therefore remarkable.