How did they know the radius of the earth in ancient times? We've all heard how in ancient times they determined that the earth was round, and were able to calculate its radius based on the measured shadows of two separate sticks in the ground at noon.  I am not questioning the math used, but instead the time constant and means of communication between the locations.
Specifically, how would the person monitoring the experiment at stick "A" communicate that it was high noon there so that the person at stick "B" could note direction and length of the shadow at that location?  It's not like they had precision timepieces that in the absence of a means to communicate wirelessly they could synchronize prior to separation.
Might they have been close enough to send a visual signal at the appropriate time?  I presumed more distance might be needed for any measurable difference in the shadows.
 A: You can tell local solar noon with a sundial, no communication needed. If the locations are at the same longitude, that time is the same, and the distance between them represents the length of the arc whose angle the shadows measure.
A: Synchronization of measurements or precision time pieces are not required. The local time when the measurements were taken is what is relevant.
You are also referring to Eratosthenes's experiment.
In summary from above link:
Considering two cities along the same meridian and measuring both the distance between them and the difference in angles of the shadows cast by the sun on a vertical rod (a gnomon) in each city at noon on the summer solstice. The two cities used were Alexandria and Syene (modern Aswan), and the distance between the cities was measured by professional bematists.
So you don't actually need the measurements to be synchronized. You just need to be at two specific locations (along the same longitude) at noon on separate days, then observe the angles cast by the shadows.
You may also be interested in reading about Al-Biruni who was a Persian mathematician who used triangulation to measure the radius of the earth, and did this at about the end of the first millennium 1,000 AD. His calculation of earth's radius $R_E\approx 6339.6\ km$ is approximately within $0.27\%$ error of the known radius.
