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Archimedes` constant PI and the Square Root of 3
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In the 3rd century B.C. Archimedes considered and circumscribed
polygons of 96 sides and deduced that 3 + 10/71 < PI < 3 + 1/7 .One of the most
frequently discussed questions in the history of mathematics is the "mysterious" approximation of
sqrt(3) used by Archimedes in his computation of pi.
"...the calculation [of pi] starts from a greater and lesser limit to the value of sqrt
(3), which Archimedes assumes without remark as known, namely (265/153) < sqrt(3) <
(1351/780). How did Archimedes arrive at this particular approximation? No puzzle has
exercised more fascination upon writers interested in the history of mathematics...The
simplest supposition is certainly [see Kline below].Another suggestion...is that the
successive solutions in integers of the equations x^-3y^2=1 and x^2-3y^2=-2 may have
been found...in a similar way to...the Pythagoreans.The rest of the suggestions
amount for the most part to the use the method of continued fractions more or less
disguised."
T. Heath, A History of Greek Mathematics, 1921. |
The square root of 3 is
the irrational
number,
 (Sloane's
A002194),
which has the simple periodic continued fraction
[1, 1, 2, 1, 2, 1, 2, ...] (Sloane's A040001).
Here is pi series:
For more on the next continued fraction below, see An Elegant Continued Fraction for Pi by L J Large in
American Mathematical Monthly vol 106, May 1999, pages 456-8.
We research the connection between the value of sqrt (3) and the value of pi...
We use the equation :
If we assume x=squart(3), we get:
or: 
and:
Pythagoras' Constant, the Square Root of 2, is related to the Archimedes` Constant pi,
as shown below :
or
So, we get:
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