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Comment by xattt | Hacker News Reader

Given the large distance between the cities, how did they communicate to confirm that the sun was indeed overhead at particular time and date?

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Well, the time is obviously noon by definition(*), and the date would have been well known as it's entirely predictable. Syene is about a degree or so north of the Tropic of Cancer, so essentially the only date the sun is in the zenith is the summer solstice, and that’s the day they should measure shadow length in Alexandria. Had it been farther to the south, there would have been two such days, still well known and understood by the people of the time, and had been for millennia, and it wouldn’t matter which of the days you’d pick.

(*) This was 2000 years before time zones became a thing and local solar times were disengaged from the wallclock time. Not that ancient Greeks had wallclocks.

You don't need to communicate in real time. You just need to agree on which date to do the measurement and that is determined by reference to the stars. Priests had been maintaining the calendar already for quite some time. Then you just wait for the sun to reach the zenith, noon. Now you can measure the angle of the sun from the vertical then it's relatively simple trigonometry to calculate the circumference.

See https://en.wikipedia.org/wiki/Earth%27s_circumference

The key is that this phenomenon was well known and predictable. First get the date right, then wait until local noon for the sun to be at its maximum altitude. The Egyptians had long had excellent calendars, so they could use past records to predict the date.

They measured something at solar noon. This can be independently measured without communication and only requires to agree on the date.

The key, really, was to be able to pick 2 locations on the same longitude and to have enough math/trigo to perform the calculation.

Note that they don't actually need to be on the same longitude. It just makes measuring the north/south distance between them a lot easier.

The angle between the sun and the zenith at local solar noon will be the same everywhere at a given latitude so that part doesn't care if the cities are on the same longitude..

You do need to know the north/south distance between the latitudes of the two cities, and picking two cities on the same longitude makes that easier to measure: just go straight from one to the other and note how far you traveled.

If the longitude was substantially different you'd have to use spherical trigonometry to figure out the north/south distance from the distance and bearing of the straight route between the cities, and for that you need to know the size of the sphere you are on.

Instead you'd have to do something like travel north from the southern city until you are at the same latitude as the other city (probably determined by observing the altitude of the North star), note how far you've traveled, then travel east or west along a line of constant latitude to try to reach the other city. If you miss the other city because you didn't get the latitude quite right, you'd have to move north or south, updating your north/south distance estimate, and try again, repeating until you actually hit the other city.

Thank you! I've wondered that, can't wait to see what he answer is.

I don't know whether this is true, but I think I've heard it said in connection with this story.

Egypt had a solar based-calendar, so (to a decent approximation) on every named date the sun would be in the same place in the sky.

So all that would be needed was for it be known that the sun shined straight down the well on (for instance) the 14th day of the 2nd Month of Growth (I had to look up the Egyptian calendar to get that date!), and Eratosthenes just needed to measure on that date.

The trickiest part was to know the effective distance component in the North/South direction. Luckily the Nile is roughly oriented that way anyway so the distance between the two cities is roughly the same as the North/South component

The communication requirements are minimal. The only real requirement is that we agree on which day to perform the measurement.

You and I could sit at Christmas dinner, and conspire to measure the shadow of a 1metre pole at solar noon on Easter Sunday, at our respective locations.

We could then compare those measurements when we meet next Christmas.

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