## History and applications

### Pi as a limit

The mathematician François Vieta (1540–1603) gave the first theoretically precise expression for \(\pi\), known as Vieta's formula:

\[ \dfrac{2}{\pi} = \sqrt{\dfrac{1}{2}} \times \sqrt{\dfrac{1}{2}+\dfrac{1}{2}\sqrt{\dfrac{1}{2}}} \times \sqrt{\dfrac{1}{2}+\dfrac{1}{2}\sqrt{\dfrac{1}{2}+\dfrac{1}{2}\sqrt{\dfrac{1}{2}}}} \times \dotsb. \]This expresses \(\pi\) as the limit of an infinite product.

John Wallis (1616–1703), who was one of the most influential mathematicians in England in the time just prior to Newton, is also known for the following very beautiful infinite product formula for \(\pi\):

\[ \dfrac{\pi}{2} = \dfrac{2\times 2\times 4\times 4\times 6\times 6\times \dotsb}{1\times 3\times 3\times 5\times 5\times 7\times \dotsb}. \]He also introduced the symbol \(\infty\) into mathematics.