astronomy - How did Eratosthenes determine that Alexandria and Syene . . . We assume first that Alexandria and Syene are on the same meridian Cleomedes does not give a detailed account of the method, that would be pointless, Eratosthenes' work still being available, but a pedagogical précis meant for readers scared of the complex geometric arguments of the original work
Is it known where Eratosthenes first heard of the well of Syene? Eratosthenes had heard from travelers about a well in Syene (now Aswan, Egypt) with an interesting property: at noon on the summer solstice, which occurs about June 21 every year, the sun illuminated the entire bottom of this well, without casting any shadows, indicating that the sun was directly overhead
How did Eratosthenes knew the exact time of the day? At Syene the longitude of the North South showed no shadow on the solstice, since the tilt of the earth is 23 5° and Syene's latitude is 24° An assistant in Alexandria could have made the same measurement when the East West shadow was 0, and came up with 7 2° North South There was no need to make simultaneous time measurements
geography - Circular reasoning in Eratostheness measurement of the . . . I remember that I read, some time ago, that Erathostenes calculated the circumfernce of the Earth using the distance between Syene and Alexandria, calculated in turn by others starting from the circumference of the Earth
Why did Columbus think the Earth was much smaller than it is? Cleomedes assumes that the distance between Syene and Alexandria was 5,000 stadia (a figure that was checked yearly by professional bematists, mensores regii) 6 He assumes the simplified (but false) hypothesis that Syene was precisely on the Tropic of Cancer, saying that at local noon on the summer solstice the Sun was directly overhead
How did Eratosthenes know the Sun was very far away? Eratosthenes calculated the radius of the Earth from the difference of the lengths of shadows between Aswan and Alexandria were different (see also here) But this could also happen if the Earth w
Historical knowledge of Distance of Earth from Sun The best answer I can think of is to refer you to a 1985 book by Albert van Helden, "Measuring the Universe, Cosmic Dimensions from Aristarchus to Halley" (1985, Univ of Chicago Press) The overall history is intricate, it would not be useful for me to try to summarize anything of it: just read the book Van Helden does refer to what little is known of Eratosthenes' work in this connection It
What are the wrong assumptions made in Anagaxoras model in his . . . On the other hand, Alexandria is located $800$ km ($=5000$ stadia) northwards from Syene In this configuration, Anaxagoras has measured an angle of $7,2°$ between the vertical in Alexandria and the direction of the sunrays shining in Alexandria