That the ratio of the
circumference to the diameter of a circle is constant (namely, pi) has been
recognized for as long as we have written records.
A ratio of 3:1 appears
in the following biblical verse:
And he made a molten
sea, ten cubits from the one brim to the other: it was round all about, and his
height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23; II Chronicles 4, 2.)
The ancient Babylonians
generally calculated the area of a circle by taking 3 times the square of its
radius (π=3), but one Old Babylonian
tablet (from ca. 1900-1680 BCE) indicates a value of 3.125 for pi.
Ancient Egyptians
calculated the area of a circle by the following formula (where d is the diameter of the circle):
[(8d)/9]power 2
This yields an
approximate value of 3.1605 for pi.
The first theoretical
calculation of a value of pi was that of Archimedes of Syracuse (287-212 BCE),
one of the most brilliant mathematicians of the ancient world. Archimedes
worked out that 223/71 < π < 22/7. Archimedes's results rested upon approximating the area
of a circle based on the area of a regular polygon inscribed within the circle
and the area of a regular polygon within which the circle was circumscribed.
Beginning with a
hexagon, he worked all the way up to a ploygon with 96 sides!
European mathematicians
in the early modern period developed new arithmetical formulae to approximate
the value of pi, such as that of James Gregory (1638-1675), which was taken up
by Leibniz:
π/4= 1 - 1/3 + 1/5 - 1/7 + . . . . . . . . . . .
One problem with using
this formula to calculate the value of pi is that you would have to add 5
million terms to work out a value of 
/4 that extends to 6 or 7 decimal places!
/4 that extends to 6 or 7 decimal places!
In 1706, another
mathematician named John Machin developed a refinement on Gregory's formula,
yielding the formula still used today by computer programmers to compute pi:
𝛑/4=4arctan(1/5)-arctan(1/239)
Using this formula, an
Englishman named William Shanks calculated pi to 707 places, a labor of many
years, which he published in 1873. (Only 527 places were correct, however!)
An eighteenth-century
French mathematician named Georges Buffon devised a way to calculate pi based
on probability. Buffon's method begins with a uniform grid of parallel lines, a
unit distance apart. If you drop a needle of length k < 1 on the grid, theprobability that the needle falls across a line is 2k/𝛑. Various people have tried
to calculate pi by throwing needles. Depending on when you stop the experiment,
you can obtain a reasonably accurate estimate of pi.
was introduced by the
British mathematician William Jones in 1706, who wrote:
3.14159 =𝛑
This symbol was adopted
by Euler in 1737 and became the standard symbol for pi.
Some people are just crazy about pi!
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