Viète's formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing the mathematical constant π:
It is named after François Viète (1540–1603), who published it in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII.
At the time Viète published his formula, methods for approximating π to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of Archimedes of approximating the area of a circle by that of a many-sided polygon, used by Archimedes to find the approximation
However, by publishing his method as a mathematical formula, Viète formulated the first instance of an infinite product known in mathematics, and the first example of an explicit formula for the exact value of π. As the first formula representing a number as the result of an infinite process rather than of a finite calculation, Viète's formula has been noted as the beginning of mathematical analysis and even more broadly as "the dawn of modern mathematics".
Using his formula, Viète calculated π to an accuracy of nine decimal digits. However, this was not the most accurate approximation to π known at the time, as the Persian mathematician Jamshīd al-Kāshī had calculated π to an accuracy of nine sexagesimal digits and 16 decimal digits in 1424. Not long after Viète published his formula, Ludolph van Ceulen used a closely related method to calculate 35 digits of π, which were published only after van Ceulen's death in 1610.
Viète's formula may be rewritten and understood as a limit expression
where an = √2 + an − 1, with initial condition a1 = √2. Viète did his work long before the concepts of limits and rigorous proofs of convergence were developed in mathematics; the first proof that this limit exists was not given until the work of Ferdinand Rudio in 1891.
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