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Mathematical economics is old enough to be respectable, but not all economists respect it. It has powerful supporters and impressive testimonials, yet many capable economists deny that mathematics, except as a shorthand or expository device, can be applied to economic reasoning. There have even been rumors that mathematics is used in economics (and in other social sciences) either for the deliberate purpose of mystification or to confer dignity upon common places as French was once used in diplomatic communications.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
Eratosthenes and the Mystery of the Stades
How Long Is a Stade?
After seeing Eratosthenes’ brilliant argument that the Earth’s circumference is 250,000 stades, one naturally asks, “ What is the length of a stade?” Unfortunately, this question has no simple answer. Without an International Bureau of Standards to ensure consistency of weights and measures throughout the ancient world, it is very likely that measures such as the stade varied slightly from region to region [2, p.46]. Scholars disagree greatly on the extent to which the stade may have varied in the ancient world. Scholar of Greek antiquity Carl Friedrich Lehmann-Haupt claims the existence of at least six different stades [2, p.43 ]. To the contrary, astronomer and historian Dennis Rawlins makes the following claim.
While the assertions of these two men represent the opposing extremes in this debate, there is an array of theories which lie somewhere in between. A common approach to this mystery is to examine the stade’s relationship to other ancient units of length.
Book Two in The Histories by the ancient historian Herodotus (480-425 BCE) tells us that 1 stade is equal to 600 Greek feet. Like the stade, the Greek foot exhibits some regional variation. However, all instances of the Greek foot appear to conform roughly to one of three basic lengths. To distinguish between these variations, scholar of Greek architecture Burkhardt Wesenberg refers to them as the “Attic” (from Asia Minor and southern Italy), the “Doric” (from Greece and Sicily), and the “Ionic” (used throughout the Greek civilization). Each of these variations of the Greek foot, when multiplied by 600, yields a stade length that corresponds closely to one of the six claimed by Lehmann-Haupt [7, pp.359-360 ]. Such correspondence lends credence to the argument that there was more than one stade used in the ancient world, and furthermore, that one of these stades may have been used by Eratosthenes.
The 185 meter stade, as claimed by Rawlins earlier, is the most commonly accepted value for the length of the stade used by Eratosthenes in his measurements of the Earth. This is so because a great number of authors from the first century CE onward make reference to the fact that 1 Roman mile is equal to 8 stades. History tells us that the Roman mile is equal to 5000 Roman feet, each of which is just short of the familiar English foot. The exact difference between the Roman foot and the English foot is uncertain, but if 1 Roman foot is taken to be approximately 11.65 English inches, then one Roman mile is approximately equal to 1479 meters. Taking 1/8 of this Roman mile gives the length of 1 stade as approximately 184.8 meters. Again, this length corresponds to one of Lehmann-Haupt’s six stades. He refers to this most frequently accepted stade as the “Italian” stade [2, pp.42-44 ].
By examining the relationship between the stade, the Greek foot, and the Roman mile, four distinct stade lengths are obtained. Using the names provided by Wesenberg and Lehmann-Haupt, each of the four stades is listed in ascending order along with the corresponding Greek foot.
Using these four stades, modern approximations of Eratosthenes’ 250,000 stades can be obtained. Below, the modern equivalent of 250,000 stades is given for each type of stade. Also given is the percent difference from the modern accepted value for the equatorial circumference of the Earth, which is approximately 40,075 kilometers [21 ].