How could Eratosthenes measure the circumference of the Earth?

Question

Some 2,200 years ago, Eratosthenes calculated the radius of the Earth.

A brief recap

Plant a stick in the ground vertically, and wait until the sun is directly above the stick, that is until there is no shadow cast (or as is said to be the case historically, stand at the bottom of a very deep well in Syene and see the sun perfectly above you).

If at the same time, a friend plants a second stick in the ground in Alexandria (which was known to be about 800km from Syene), and then measures the length of the shadow, one could calculate the circumference of the Earth.

See also Cosmos: Sagan on Eratosthenes' calculation of Earth's Circumference

The problem I have never been able to figure out is how did Eratosthenes and his "theoretical friend" in Seyene know they were measuring the length of the shadow at the same time?

In fact, how could Eratosthenes prove that any two measurements of the sun happened "at the same time", when the primary way time was measured at the time was based on the sun?

Answer 1

He synchronised them to the solar zenith.

Eratosthenes knew that on the day of the summer solstice, the sun passed vertically above Syene, which lies very close to the Tropics of Cancer. As the traditional account goes, the sun was directly above a vertical well at Syene, whereas at Alexandria the columns of the Library always leaves a shadow. Either way, he concluded that the sun was 7.2° (1/50) from vertical at Alexandria at its peak.

By making his observations when the sun was highest on the day of the summer solstice, he could be reasonably confident that it was essentially "at the exact same time" to all his practical intents and purposes.

Answer 2

Simple.

While the earth moves around the year, the sun seemingly moves around between the Tropic of Cancer (north) and the Tropic of the Capricorn (south). In the north this is begin of summer and the sun reaches the highest point.

The first city where the deep well exists is the city of Syene (now Assuan) which is almost exactly on the Tropic of Cancer, so at almost exactly on June, 21th the sun is almost exactly vertical and the well is lit.

Knowing this, Erastothenes only needed to measure the angle of the sun in Alexandria during June, 21th and knowing that Syene is vertical gets the difference easily. He only need to know the distance between Alexandria and Syene (which must be measured), multiply it with 360 degrees / difference and get the Earth diameter.

Quite genial.

ADDITION: It was asked if it was not necessary to have both cities on the same meridian to get the correct result. While in fact not being on the same meridian does induce an error, this error is neglible because the zenith point of the sun moves with 397 m/s at Alexandria (423 m/s at Syene) in west-east direction. This is supersonic; so even a bigger west-east offset is covered in a short time and an alignment is not needed.

While not necessary, it is quite easy to get the exact geographical north-south direction. Put a straight stick vertical into the earth and draw a circle around it so that the shadow will cross it early and late. Mark the exact points where the shadow touches the circle. Now draw two circles around the points so that the circles intersect. The two intersection points are forming a line which is exactly north-south ! If you do that with stars and a level wall, this will even outperform GPS in precision.