Eratosthenes in 250 BC determined that the Sun is in Siena (now Aswan) at the zenith at the moment when in Alexandria

Eratosthenes in 250 BC determined that the Sun is in Siena (now Aswan) at the zenith at the moment when in Alexandria it is located 7 ° 12 ‘south of the zenith. The distance between observation points was 5000 stages. Determine what one stage is.

the distance between Siena and Alexandria on the surface of the Earth is Ry, where R is the radius of the Earth, and y is the zenith distance of the Sun in Alexandria, when it is at its zenith in Siena, expressed in radians (the distance to the Sun is much larger than the dimensions of the Earth, and the direction to the Sun of the two cities can be considered parallel). Substituting the numerical values, we find that 5000 stages separating the two cities are equal to 800 km, therefore, one stage was equal to 160 meters.

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