Eratosthenes observed that during noon time of summer solstice the Sun’s rays cast no shadow in the city Syne which was located 500 miles away from Alexandria. At the same day and same time he found that in Alexandria the Sun’s rays made 7.2 degree with local vertical. He realized that this difference of 7.2 degree was due to the curvature of the Earth.
The angle 7.2 degree is equivalent to \(\frac{1}{8}\) radian. so, θ = \(\frac{1}{8}\) rad;
If S is the length of the arc between the cities of Syne and Alexandria, and if R is radius of Earth, then
S = Rθ = 500 miles,
so radius of the Earth
R = \(\frac{500}{θ}\) miles; R = 500 \(\frac{miles}{\frac{1}{8}}\)
R = 4000 miles
1 mile is equal to 1.609 km. So, he measured the radius of the Earth to be equal to R = 6436 km, which is amazingly close to the correct value of 6378 km.
