At the time Viète published his formula ( in 1593/ Variorum de rebus mathematicis responsorum, liber VIII), methods for approximating π to 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.
References:http://www.theoremoftheday.org/GeometryAndTrigonometry/Viete/TotDViete.pdf
https://www.revolvy.com/main/index.php?s=Vi%C3%A8te%27s%20formula&item_type=topic
Animation by Cliff Pickover / IBM researcher
Corina Marinescu
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