**Ludolph van Ceulen**in 1596 using the polygon method, first came to calculate 20 decimal digits, then 35. Van Ceulen wasn't the last to use the method: for example

**Willebrord Snellius**in 1621 calculated 34 digits, while the Austrian astronomer

**Christoph Grienberger**in 1630 reached a record 38 digits using a 1040-sided polygon: this result is the most accurate ever achieved using the polygon method.

The infinite series supplanted this method: the first to use them in Europe was the French mathematician

**François Viète**in 1593 \[\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots\] And in 1655

**John Wallis**\[\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\] European mathematics, however, had come to this method only after Indian mathematics, albeit independently. In India, in fact, there is evidence of first approaches of this kind between 1400 and 1500. The first infinite series used to calculate \(\pi\) is found, in fact, on the pages of the

*Tantrasamgraha*(literally "

*compilation of systems*") of the Indian astronomer

**Nilakantha Somayaji**, circa 1500-1501. The series, presented without any proof (later published in the

*Yuktibhāṣā*, circa 1530), was attributed by Nilakantha to the mathematician

**Madhava of Sangamagrama**, who lived between 1350 and 1425 circa. Apparently Madhava discovered several infinite series, including many that contain the sine, cosine, and tangent. The Indian mathematician used these series to reach up to 11 digits around 1400, a value that was improved around 1430 by the Persian mathematician

**Jamshīd al-Kāshī**using the polygon method.

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