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Around 250 B.C., the Greek mathematician Archimedes calculated
the ratio of a circle's circumference to its diameter. A
precise determination of pi, as we know this ratio today, had
long been of interest to the ancient Greeks, who strove for
precise mathematical proportions in their architecture, music,
and other art forms.
In Archimedes' day, close approximations of pi had been known
for over 1,000 years. An Egyptian document dated to 1650 B.C.,
for example, gives a value of 4 (8/9)2, or 3.1605.
Archimedes' value, however, was not only more accurate, it was
the first theoretical, rather than measured, calculation of
pi.
How did he do it? The
interactive at left
illustrates Archimedes' basic approach. It finds an
approximation of pi by determining the length of the perimeter
of a polygon inscribed within a circle (which is less than the
circumference of the circle) and the perimeter of a polygon
circumscribed outside a circle (which is greater than the
circumference). The value of pi lies between those two
lengths.
By doubling the number of sides of the hexagon to a 12-sided
polygon, then a 24-sided polygon, and finally 48- and 96-sided
polygons, Archimedes was able to bring the two perimeters ever
closer in length to the circumference of the circle and
thereby come up with his approximation.
Specifically, he determined that pi was less than 3 1/7 but
greater than 3 10/71. In the decimal notation we use today,
this translates to 3.1429 to 3.1408. That's pretty close to
the known value of 3.1416. (For simplicity's sake, we round
off all figures to four decimal places.)
Like Archimedes' approach, our interactive doesn't rely on
specific measurements. The diameter of the circle is given an
arbitrary value of 1; it doesn't matter if that number
represents an inch, a foot, or a light-year. Also like
Archimedes' approach, the interactive determines the length of
a side of each triangle, relative to the diameter, based on
the angle opposing the side being measured.
Our interactive differs from Archimedes' approach in three key
ways, however. First, it makes use of algebra and modern
trigonometry, which were unknown in Archimedes'
day—Archimedes used geometry instead. For example, he
knew the ratio between one line and another in certain
triangles and with this knowledge was able to figure out the
length of the perimeter of a hexagon.
Second, we use decimal notation, which wasn't invented until
hundreds of years after Archimedes' death. To work with
non-whole numbers, the ancients relied on ratios. Any
calculator will tell you that the square root of 3 is 1.7321.
For Archimedes, that value was 265/153 (which equals 1.7320 in
decimal notation).
Finally, our interactive increases the number of sides of the
hexagon to 96 by increments of 1 rather than by the doublings
Archimedes used. The idea is to give you a clearer sense of
how ever closer in length to the perimeter of the circle the
length of the hexagonal perimeters becomes with each added
side.
It is interesting to note that even today pi cannot be
calculated precisely—there are no two whole numbers that
can make a ratio equal to pi. Mathematicians find a closer
approximation every year—in 2002, for example, experts
at the University of Tokyo Information Technology Center
determined the value of pi to over one trillion decimal
places. But this is academic: the value determined by
Archimedes over 2,000 years ago is sufficient for most uses
today. —Rick Groleau
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