what are some of the oldest and most accurate approximations of pi? I read that there is a tradition in the Jewish literature of an approximation of pi given in the prophets was very accurate ($\frac{3\times111}{106} \approx 3.14150\ldots$ - difference of about $.0026\%$)
I'm curious what is the next ancient (>1000 years) more accurate approximation of Pi?  
 A: According to [wikipedia] the approximation $\frac{22}{7}=3.\overline{142857}$ is known since antiquity, and Archimedes proved it is smaller than $\pi$.
Note that you can obtain  good rational approximations of $\pi$ by using the continued fraction method. write:
$$\pi=3+\{\pi\}=3+\frac{1}{\frac{1}{\{\pi\}}}=3+\frac{1}{7+\{\frac{1}{\{\pi\}}\}}$$
Up to now we have the approximation $\frac{22}{7}$ But we can take it a step further:
$$3+\frac{1}{7+\{\frac{1}{\{\pi\}}\}}=3+\frac{1}{7+\frac{1}{\frac{1}{\{\frac{1}{\{\pi\}}\}}}}=3+\frac{1}{7+\frac{1}{15+\{\frac{1}{\{\frac{1}{\{\pi\}}\}}\}}}\}$$
Which gives us the approximation $\frac{333}{106}=3.1\overline{4150943396226}$
The next approximation would be $\frac{355}{113}$ and after that we have $\frac{103993}{33102}=3.1\overline{415926530119026040722614947737296840070086399613316}$

Now that I look at it the fraction you gave is $\frac{333}{106}$, which is also one of the convergents of $\pi$. It can be proven that the best approximations (The ones that give the best size of denominator vs accuracy ratio are all convergents.)
A: From the link given by Brian, Ptolemy obtained $\pi\approx \frac{377}{120}$ in the 2nd century. This is closer to $\pi$ than $\frac{333}{106}$.
$$\frac{377}{120}-\pi \approx 7.4·10^{-5}$$ 
$$\pi-\frac{333}{106} \approx 8.3·10^{-5}$$
The error has an integral formula similar to the one given by Dalzell for $\frac{22}{7}-\pi$ (see Series and integrals for inequalities and approximations to $\pi$), as well as a corresponding series with constant numerator terms.
$$\begin{align}
\frac{377}{120}-\pi
&=
\frac{1}{2}\int_0^1 \frac{x^5(1-x)^6}{1+x^2}dx \\
&=\sum_{k=1}^\infty \frac{7!}{(4k+2)(4k+3)(4k+4)(4k+5)(4k+7)(4k+8)(4k+9)(4k+10)}
\end{align}$$
Ptolemy's approximation is the mediant fraction of $\frac{22}{7}$, given by Archimedes, and $\frac{355}{113}$, from Zu Chongzhi.
$$\frac{22+355}{7+113}=\frac{377}{120}$$
The latter is closer to $\pi$, but it is dated about three centuries later.
