earth's circumference calculated by Eratosthenes.

[duccio]

I heard a report on the radio that said that the ancient Greeks had proven that the Earth was a sphere, and had closely calculated it's circumference. (About 1800 years later, Columbus decided to test the theory.) I experimented with the method described on the radio which used a vertical, meter long pole stuck in the ground in two locations roughly N. and S. of each other - though I did my experiment on paper with a simple drawing and imagination. I think their method works and the simplicity of their logic is really sweet. (HINT: A compass would make things easier - they used the sun.)

Let's say the ancient Greeks came to Portland and San Francisco and put their meter high sticks into the ground in those places, and knew that Portland was 537 miles north of SF (or about 3 sailing days by bireme @ 8 mph, or 27 days walking 20 miles/day...) How did they do their calculations to come up with a circumference of 25,000 miles?

I'll reply with the solution later in a comment, maybe.



Solution: If the poles are placed vertically in the ground, at noon their shadows will point due north and also be their shortest length during the day. At noon the angle of the top of this shadow triangle is measured. It so happens that the angle at Portland is 45.3° while that at SF is 37.5°. The difference between these two measurements is 7.8°. 7.8° is the portion of the 360° circumference of the earth lying between Portland and SF. The exact distance is 537 miles, so it is a proportional problem to calculate the full 360° distance.
7.8/537 : 360/C
7.8C = 537x360 = 193320
193320/7.8 = C = 24,784 miles