Archimedes' Approximation of Pi
one of the major contributions Archimedes made to mathematics was his method for
approximating the value of pi. It had long been recognized that the ratio of the circumference of
a circle to its diameter was constant, and a number of approximations had been given up to that
point in time by the Babylonians, Egyptians, and even the Chinese. There are some authors who
claim that a biblical passage
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also implies an approximate value of 3 (and in fact
there is an interesting story
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associated with that).
At any rate, the method used by Archimedes differs from earlier approximations in a
fundamental way. Earlier schemes for approximating pi simply gave an approximate value,
usually based on comparing the area or perimeter of a certain polygon with that of a circle.
Archimedes' method is new in that it is an iterative process, whereby one can get as accurate an
approximation as desired by repeating the process, using the previous estimate of pi to obtain a
new one. This is a new feature of Greek mathematics, although it has an ancient tradition among
the Chinese in their methods for approximating square roots.
Archimedes' method, as he did it originally, skips over a lot of computational steps, and is not
fully explained, so authors of history of math books have often presented slight variations on his
method to make it easier to follow. Here we will try to stick to the original as much as possible,
following essentially Heath's translation
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.
The Approximation of Pi
The method of Archimedes involves approximating pi by the perimeters of polygons inscribed
and circumscribed about a given circle. Rather than trying to measure the polygons one at a time,
Archimedes uses a theorem of Euclid to develop a numerical procedure for calculating the
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 Spring '11
 sarpca
 Archimedes

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