View Full Version : The Real Science of Eratosthenes

06-23-2017, 03:02 PM

In the 1760’s Charles Mason and Jeremiah Dixon surveyed a line of the same name. They found larger than normal systematic errors, which Henry Cavendish attributed to the gravitational pull of the Allegheny Mountains. Several years later Cavendish did a series of sensitive experiments to determine the gravitational constant, which is in every physics textbook. The journey from a line in the sand to the event horizon of a black hole started more the 2,000 years ago.

The Real Science of Eratosthenes (276-194 BC)

The connection between observation, logic, and calculation couldn’t be better illustrated than by Eratothenes’s measurement of the circumference of the Earth.

We need a reference time and place; for this document it will be:
Toronto, Canada, latitude: 48 degrees, 42 minutes.
Summer Solstice, 2017, Wednesday June 21
Warning; Solar noontime (1:19 PM) is required when shadows fall exactly north.

Only two measurements are required, a stick and the length of its shadow.

Eratosthenes’s Method

Assume that the yellow rays of sunlight are parallel. One line strikes the equator and there is no shadow. Farther north, the black line casts a shadow. The black line crosses two parallel lines so that the shaded angles are congruent to each other.


The Tropic of Cancer is about 23.4 degrees north. The distance to the equator is 1617 miles. Circumference of the Earth is
1617 x 360 / 23.4 = 24,877 miles

Data and Calculations

We need the distance between Toronto and the tropic of Cancer,
43’ 42” – 23’ 27” = 20.25 degrees. Since there are 69.1 miles per degree the distance is 1400 miles.

Stick: 115.1 cm
Shadow: 42.7 cm

Tangent is 42.7 / 115.1 = 0.371
Angle is arctangent (0.3695652) = 20.354 degrees

Circumference is distance x 360 / 20.354
= 1400 x 17.687
= 24,762 miles which is about 100 miles less than the meridional circumference.


Why do we see difference values for the Earth’s equatorial circumference on the internet?

This value is given by Wikipedia, <Earth> 40 075.017 kilometers, and since 1 km = 0.621371 mile we have, circumference = 24,901.45 miles.

If we use 1 km = 0.621 the value is 24,886.59 which is off by about 14 miles.

News Headline Oct 1, 1999: NASA lost its $125 –million Mars Climate Orbiter because of conversion from metric to imperial units. Whoops!