DIY: Measure the Circumference of Earth.
(From https://en.wikipedia.org/wiki/Spherical_Earth)
Strategy
It is known that in Syene, Egypt, at a certain date and time, the sun is directly overhead. Vertical objects cast no shadows, and the sun shines straight to the bottom of wells.
At the same moment, in Alexandria, close to 5000 stadia (about 500 miles) away, vertical objects cast shadows. By using such shadows to measure the angle a of the Sun's rays off the vertical, one can calculate the circumference C of Earth by assuming
(500 miles)/C = a/360°
In other words, 500 miles is the same fraction of the circumference C as the angle a is a fraction of a full circle, or 360°. Therefore,
C = (360°/a) x 500 miles
But how does one measure the Sun's angle accurately?
To one end of a straight rod several meters in length, attach a string, and to its other end, a stone. Stand the rod upright in a hole, and make it vertical using the hanging stone as a guide. Pack the sand (this is Egypt, so what else?) tightly around the rod, keeping it vertical. Mark the rod where it enters the ground (point O).
At the time when one knows that the sun is straight overhead in Syene, mark in the sand the rod's shadow in Alexandria (line O-S). Remove the rod and lay it perpendicular to the shadow's mark, aligning the rod's point O with the hole. In the sand, draw the line S-T where T is the top of the rod. Use a compass to measure the angle STO, which is angle a. (Note: angle SOT is a right angle.)
drawing coming
This angle turns out to be about 7°, or about one-fiftieth (1/50) of 360°. Thus 500 miles is about 1/50 of the circumference C, or C is 50 x 500 = 25,000 miles.
Not bad. The modern figure is 24,901 miles.
[ About significant figures and precision: Probably the best you can do at measuring the angle a is to the nearest whole degree, or at the most optimistic, the nearest two or three tenths of a degree. So the angle will have at most 2 significant figures (say 7.3 ± 0.2°) The measured distance between the cities is probably no better than ± 10 miles (it was paced off by servants, they say.) ]. So the stated distance of 25,000 stadia or 25,000 miles is about as precise as one could be. ]
