## Sunday, May 4, 2014

### Geometrical Units and the Size of the Earth

This is a convenient time to consider the proportion between the unit sphere and the dimensions of the Earth. How do we convert the angular units into units of length? When the meter was defined it was assumed that the distance from the equator to the poles was 10,000 km. So we can define an angle to length conversion factor, α2λ = 10000 km/90°, and see what we get in terms of the customary civil units of measurement.

This looks like someone was really clever in defining the nautical mile and the foot but one can do the same thing with the ancient Greek units of measure, the pous, the stadion and the plethron.

The Greek mathematician, Eratosthenes, may have noticed this in ancient times and his rough measurements may have used geometrical units. If this is true then the ancient estimate of the size of the Earth was accurate to about 3% while that of the more modern foot has an accuracy of about 1%. It should be noted that a pous = 300 mm was used by the Phoenicians and the ancient Egyptians.