How could Eratosthenes measure the circumference of the Earth?

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.

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?

• Migrate to history of science? Aug 5 '15 at 9:14
• @SamuelRussell They have a StackExchange for that now? I'm surprised, that subject seems awfully specific. Aug 5 '15 at 9:30
• @IQAndreas I also think they are going more and more specific and that this makes no sense
– o0'.
Aug 5 '15 at 9:39
• @SamuelRussell it's still in-topic here, migrate only if it gets no answers here.
– o0'.
Aug 5 '15 at 9:39
• This question has received multiple answers. If they haven't answered your question, it would be helpful to explain why. Otherwise, please consider marking one accepted. If you have further follow-up questions, you should ask them in a separate post. Sep 5 '15 at 20:36

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.

• Would that not require Alexandria to be exactly north of Syene (with any deviation causing a skew in the result)? Moreover, that would also require the "magnetic north" (assuming they used compasses) to match up with the "rotational axis north", right? Are both those true, or at least close enough for these measurements? Aug 5 '15 at 12:02
• @IQAndreas Not really; the difference is enough for him to get a rough estimate of the earth's circumference (in contrast to the earth being flat). No one is claiming that Eratosthenes was 100% accurate. Aug 5 '15 at 12:21
• @IQAndreas: even given that Alexandria isn't exactly north of Syene (and it isn't), it's not as though it's a whole time zone away or anything. If both locations make their measurements at solar zenith, then they will not be precisely simultaneous, they'll be a few minutes apart. But the change in the angle of the sun at zenith can be observed not to vary very much in a given location from one day to the next, never mind adjusting for a few minutes, and especially so at the solstice when it is "moving" slowest. So those few minutes are not a significant source of error here. Aug 5 '15 at 12:44

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.

• Would that not require Alexandria to be exactly north of Syene (with any deviation causing a skew in the result)? Moreover, that would also require the "magnetic north" (assuming they used compasses) to match up with the "rotational axis north", right? Are both those true, or at least close enough for these measurements? Aug 5 '15 at 12:03
• @IQAndreas Deviations probably did skew the result somewhat, and according to Wikipedia's account, between that and natural limitations of "a step is about a meter" he was off by about 7000 kilometers -- about 15%, in other words. Still impressive, given the imprecise tools he had to work with. Aug 5 '15 at 12:34
• @IQAndreas: you don't need a magnetic compass to measure the sun at zenith. For example you can repeatedly put a dot on the ground at the tip of the shadow (these dots will describe a curve), and whichever one is the shortest shadow was when the sun was at zenith. Assuming the thing casting the shadow is vertical, anyway. Eratosthenes didn't have a magnetic compass, but he did have a plumbline ;-) Aug 5 '15 at 12:55
• You also don't need a magnetic compass to determine which direction north was; see Egypt, for example. The motion of the sun itself tells you the direction. So it should've been quite possible to determine how far north Alexandria was from Syrene.
– Joe
Aug 5 '15 at 18:30
• Not sure how this works taking into account the analemma (en.wikipedia.org/wiki/Analemma) of the sun. Can't vote you up until you convince me this works other than at the equinox. Aug 5 '15 at 22:40