Eratosthenes' Experiment

Using basic geometry, Greek mathematician Eratosthenes determined the circumference of the Earth within 0.4% error of today's value.

Key Points

Eratosthenes was able to calculate the circumference of the Earth through observing reflections of sunlight in wells at various locations.
Eratosthenes' calculations were based on the assumptions that rays of sunlight were parallel and that the Earth was a sphere of matter.
Eratosthenes was able to obtain his result by taking measurements, and using his data in conjunction with a mathematical equation to produce a result.

Term

circumference
The line that bounds a circle or other two-dimensional figure

Full Text

Eratosthenes' Experiment

Eratosthenes (~200 BC) worked as a librarian in the great library of Alexandria, Egypt. Those were times of greatly increasing travel from the East to West and back.

Eratosthanes sought to know the circumference of Earth. Knowing geometry and having observed eclipses of the moon, he believed Earth was a sphere of matter. To measure its circumference, he devised a method that used the rays of sunlight that hit Earth which he assumed arrive parallel.

Use Eratosthenes' data to calculate the circumference of Earth.

Wells were common in Egypt. To dig straight down, a worker would place a plank across the well opening and at at its center he would lower a mass attached to a string. By Earth's gravity, the mass hung straight down; the line of the string established a "vertical" at that geographic location.

The mass hung in one well at Syrene showed something special. At precisely noon on the solstices the shadow of that masses fell directly to the well bottom, in near-perfect alignment with the rays of sunlight.

Elsewhere at noon (on days near the summer solstice) were one to look into the well one would see the full blinding reflection of rays of sunlight. Eratosthenes reasoned that this occurred because the rays of the sun struck Earth perpendicularly. Alexandria is approximately 500 miles due north of the well at Syrene. To measure the angle of incident sunlight there Eratosthenes lowered a plumb bob suspended by a string deep into a well. The sketch shows the angle of the rays at Aswan (zero degrees at noon on the say of summer solstice). Also depicted is the "1/50^th of a circle" angle measured at Alexandria at noon on the same day.

$s = r \theta$

We seek the radius (Earth's radius) associated with a circular arc length of 500 miles with the angle of arc subtended being 1/50 th of a full circle. Placing these numbers in the equation yields:

$500 \; miles = r_{earth} \left(\frac{1}{50} circle \left(\frac{2 \pi \; radians}{circle}\right)\right)$,

$r_{earth} = 3,979 \; miles$

But the circumference of a sphere equals π times its diameter:

$C_{earth} = \pi D = \pi \left(2r_{earth}\right) = \pi \left(2\left(3979 \; miles\right)\right) = 25,000 \; miles$

Today the measured radius of Earth is taken to be 3963 miles. Thus, supposing 3963 miles is the actual radius, the percent error of Eratosthenes' measurement was:

$Percent \; Error = \left(\frac{r_{Erato} - r_{now}}{r_{now}}\right) \times 100\%$;

$Percent \; Error = \left(\frac{3979 - 3963}{3963}\right) \times 100\% \cong 0.4\%$

Thus his measurement of the Earth circumference (some 2000 years ago) was in error: Less than actual by only one-tenth of a percent. Very close, indeed!

In closing, notice that Eratosthenes "made some measurements" then entered those measurements into an equation to obtain a second, grander measurement. This is a common technique of engineering and science.

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