Definition:Pi/Historical Note

Historical Note on $\pi$ (Pi)
Every ancient society that considered circles was aware of $\pi$, although in general only as a rough approximation.

In the Old Testament, the implication is that $\pi = 3$:
 * 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 round about.
 * -- $\text I$ Kings $7 : 23$

The Egyptian scribe in the  used the approximation that the area of a circle equals the area of a square whose side is $\dfrac 8 9$ that of the diameter of the circle, leading to a value of $\pi$ of $\left({\dfrac {16} 9}\right)^2 = 3 \cdotp 16049 \ldots$

By calculating the areas of regular polygons of $96$ sides, determined that $3 \dfrac {10} {71} < \pi < 3 \dfrac {10} {70}$, that is:
 * $3 \cdotp 14085 \ldots < \pi < 3 \cdotp 142857 \ldots$

That last value:
 * $3 \cdotp 142857 \ldots$

more often given as $\dfrac {22} 7$, is commonly used in schools as a good working approximation to $\pi$.

In binary notation it has the repeating pattern:
 * $\pi \approx 11 \cdotp 00100 \, 1001 \ldots$

also found more accurate approximations still.

used $\dfrac {377} {120}$, which is approximately $3 \cdotp 14166 \, 66 \ldots$

and his son determined that:
 * $3 \cdotp 14159 \, 26 < \pi < 3 \cdotp 14159 \, 27$

calculated $\pi$ to $16$ decimal places.

It was calculated by between $1596$ and $1610$.

Improvements in trigonometric techniques allowed for better methods for calculating the digits of $\pi$.

calculated $34$ places using the same techniques that used to calculate $14$.

achieved $9$ places just by considering the geometry of the regular hexagon.

was the first to devise a formula for $\pi$, which he did in $1592$.

was next, with Wallis's Product.

devised a formula in $1666$, and devised one in $1673$.

The latter is unfortunately too inefficient to be useful.