Greek thinkers were not mere abstract philosophers. Euclid’s geometry still forms the basis of that school subject. Trigonometry yielded the properties of triangles, angles sizes and so on. Eratosthenes combined this knowledge with a few acute observations of his own, and those he learned from others.

It was generally known in knowledgeable circles (unintentional pun!) that the whole sun was reflected in the bottom of a deep well in Syene (now Aswan, of the dam) at midday every midsummer, implying that the sun was directly overhead at that time and that therefore objects cast no shadow. We name such a place as a point on the earth’s equator.

At midday on the summer solstice (i.e. at precisely the same time) Eratosthenes measured the length of a shadow cast by a tower in his home town of Alexandria (near Cairo), whose height he had measured accurately. He had the distance between here and Syene paced out accurately and the resulting measurement was close to 500 miles.

He used all these figures to calculate how much the Earth curved in 500 miles, and extrapolated from that to the curvature of a full circle 360º. 

He concluded that the Earth must be 24,000 miles in circumference.

He was almost exactly right!

Now, your task is as follows

1 What are the mathematical formulae involved?

2 How do you work out the angle of inclination at the tower in Alexandria?

3 Using the figures here, what angle of inclination did he find at Alexandria?

Details:

Eratosthenes determined that the earth’s circumference was 250,000 stadia, later revised to 252,000 stadia – or about 25,054 miles (40,320 km). The actual circumference is 24,857 miles (40,009 km) around the poles and slightly more (because of bulging due to the centrifugal force) at the equator (24,900 miles or 40,079 km).

To Eratosthenes, the earth’s diameter was 7,850 miles (12,631 km). Today’s accepted mean value is 7,918 miles (12,740 km).

Little wonder Eratosthenes is accepted as the father of geodesy – the science of earth measurement.