Thursday, February 6, 2014

Eratosthenes' Earth, 2200 BC

Eratosthenes:  What curious minds do best

More than 2,200 years ago the Greek mathematician Eratosthenes was able to calculate the circumference of the Earth by using geometry, deductive reasoning and some carefully considered assumptions among astronomers of the Hellenistic period.  It was the Greek astronomer Aristarchus that presented the earliest known model of the Earth orbiting around the Sun and his ideas would have been known to Eratosthenes and he would build upon this foundation.

If the Sun was at the center then there was a philosophical inclination to believe it was also likely to be vastly larger than its orbiting bodies such as the Earth, or its Moon.  The Greeks had other, more objective, reasons for this belief and they used this and other evidence in an attempt to determine the distance between the Sun and our Earth.  The numbers they arrived at were never close to the actual truth as their means were too rudimentary and the idea of a burning sphere warming a planet from ninety-three million miles away was likely to seem absurd.  Eratosthenes believed that whatever the actual distance turned out to be, it would be great enough for the Sun’s rays to strike the Earth parallel to one another.  This would be a crucial factor in Eratosthenes’ geometric-based calculation of the Earth’s circumference. 

Numbers off but right approach

Another belief important to Eratosthenes and other astronomers was the conclusion that the Earth was a sphere.  The evidence of this was revealed by the curve of the Earth’s shadow that was cast upon the Moon’s face during a lunar eclipse.  There were more subtle forms of evidence supporting this theory, such as a departing ship’s image descending beneath the horizon, but certainly the shape of the Earth’s form projected upon the Moon was the most dramatic.

Finding the arc's size in relation to the circle

As it happened Eratosthenes was in Egypt and he had become aware that on the first day of summer, at high noon, the Sun was directly overhead in a place named Syene, now Aswan, along the Nile.  The Sun’s rays were perpendicular to the land and they shone straight down a deep well, reflecting off the water at its normally darkened depth.  Eratosthenes was also able to determine that at the same time on the same day the Sun’s rays were at an angle of just over seven degrees from the perpendicular in Alexandria, another location in Egypt.  He calculated this by measuring the shadow cast at noon in this Mediterranean town.   Knowing that a seven plus degree angle represented 1/50th of the circumference of a circle Eratosthenes now needed to know the distance from Syene to Alexandria.  His calculation would make that length about 925 kilometers in modern measurement terms.  The actual length is closer to 800 kilometers, introducing one of a number of inaccuracies.  A related inaccuracy had to do with the precision of the determined angle in Alexandria.  The length of the arc representing the distance between the two Egyptian locations should have represented 1/48th the circumference of the Earth and not 1/50th.

Fraction of circle times arc's length gives circumference

The circumference of the Earth, according to Eratosthenes, would be 46,250 kilometers or 28,721 miles using contemporary measurement standards.   It is thought Eratosthenes used the Greek measurement of stadion, believed to approximate 185 meters.   This adds to the uncertainty of Eratosthenes’ final value for the Earth’s size.  The Earth’s actual circumference around the equator is a bit over forty thousand kilometers or 24,902 miles.  Had Eratosthenes’ calculation given 800 kilometers as the distance between Syene and Alexandria his numbers would come astonishingly close to our current measurement, with only 62 miles separating the two figures. 

A kilometer is just over half a mile

In any case, the human mind demonstrated extraordinary reasoning ability twenty-two hundred years ago, armed only with primitive tools and the most limited of inherited knowledge.  Here were individuals attempting to understand their world, not through metaphors and stories, but through objective observation of the world about them and using the tool of mathematics which would improve with time in both power and sophistication.

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