If Eratosthenes had measured the angle that the Sun made with the vertical at noon on June 22 in Alexandria to have been 14o instead of 7o,approximately what value would he have estimated for the Earth's circumference?
Eratosthenes actually measured the angle made by the Sun with the vertical at noon on June 22 in Alexandria as 7 degrees 12 minutes or about 1/50th of a circle. He postulated, based on these measurements, that the distance between Alexandria and Syene was 1/50th of the circumference of the Earth. He assumed, based on 'known' facts at that time, that the distance between these two cities is 5000 stadia (or about 925 kilometers). Thus the Earth's circumference was calculated as 50 x 5000 = 2,50,000 stadia.
If he had measured it to be 14 degrees or about 1/25th of a circle, then the circumference of the Earth would have been measured (keeping all other assumptions constant) to be 25 x 5000 = 1,25,000 stadia.
This estimate would have been half his original estimate.