1.1 Eratosthenes Measures the Earth (6 pages)

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3. Use the result of the previous exercise to find the circumference of the earth in
stades.

Unfortunately, more than one distance called "stade" was in use in the ancient world and
there is some controversy about which one was used.

Exercise

4. If the Egyptian stade of about 516.7 feet was used, compute the circumference of
the earth in miles. Find a reference book (visit the library if necessary - but the
dictionary may suffice) that has the modern measurement of the circumference of the
earth and compare the two values.

The result of your computation should be pretty close to the modern value. It is of interest
to note that this measurement was known to the learned folk of Columbus' time and is one
of the reasons he found it difficult to get financial backing for his voyage west. Indeed, if
there were no American continents and only open ocean between Europe and Asia,
Columbus very likely could not have completed the voyage in the vessels of his day. He
thought the world was much smaller than it really is.

Eratosthenes' Data:

Eratosthenes' measurement rests on two pieces of data: the distance from Alexandria to
Syene and the angle he measured at Alexandria on the summer solstice. We'll talk about
each in turn.

Exactly how Eratosthenes determined the distance from Alexandria to Syene is apparently
not known. As is almost always the case in investigating ancient events, the story has to be
pieced together from accounts in manuscripts often written hundreds of years after the
event. Daniel Boorstin, in The Discoverers, claims he was told that a camel caravan
would travel 500 stades in a day and the trip to Aswan took 10 days while David Burton, in
his History of Mathematics, tells us that the distance had been measured by a surveyor
or "bematistes" trained to walk with equal strides and count the paces.

Eratosthenes seems to have been aware of the approximate nature of the number 5000 as
the number of stades from Alexandria to Syene. For one thing the phenomena of vertical
poles casting no shadows at the summer solstice was said to occur within a region of 300
stades around Syene. For another, there is evidence that Eratosthenes modified his
computed value of 250,000 stades and to have used instead the value 252,000 stades.
Presumably he would only have done so if he thought there was some room for error in his
computed value.

The change from the "round number", 250,000, to 252,000 may seem strange until we
realize that 250,000 is a "round number" in the decimal system, but not in the Babylonian
sexagesimal system - in particular 250,000 is not evenly divisible by 60 while 252,000 is.
This is further evidence of the influence of Babylonian numerical methods on Eratosthenes'
work.

Exercises:

5. Assuming that vertical poles cast no shadows on the summer solstice for a
distance of 300 stades around Syene, find the largest and smallest possible
circumference for the Earth in stades using Eratosthenes' method. Is his assumed
value of 252,000 within the likely limits of error?

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