Erasthosenes measured the circumference of the earth on the summer solstice by observing the sun at Alexandria and Syene. Was this done on the same day in one year or on the same day in two years?
By a close reading of the wikipedia article, it says that
He knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC)...He then measured the Sun's angle of elevation at noon in Alexandria by using a vertical rod, known as a gnomon, and measuring the length of its shadow on the ground (emphasis is mine).
This reading seems to imply that he had prior knowledge that the Sun in Syene was directly overhead during the summer solstice, and then on some other summer solstice he measured the shadow at Alexandria.
Also, the distance from Syene to Alexandria was measured to be about 5,000 stadia which converts to about 625 miles. I truly doubt that somebody could travel that far and see both high noons (even today).
Conclusion: Eratosthenes made the measurements on different days.