this correction. Using this he found an approximation of to 13 decimal places of accuracy when = 75. Jamshīd al-Kāshī (Kāshānī), a Persian astronomer and mathematician, correctly computed the fractional part of 2 to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point: which gives 16 correct digits for π after the decimal point: He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 228 sides. 16th to 19th centuries In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on known as Viète's formula. The German-Dutch mathematician Ludolph van Ceulen (circa 1600) computed the first 35 decimal places of with a 262-gon. He was so proud of this accomplishment that he had them inscribed on his tombstone. In Cyclometricus (1621), Willebrord Snellius demonstrated that the perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by Christiaan Huygens in 1654. Snellius was able to obtain seven digits of from a 96-sided polygon. In 1706, John Machin used Gregory's series (the Taylor series for arctangent) and the identity to calculate 100 digits of (see below). In 1719, Thomas de Lagny used a similar identity to calculate 127 digits (of which 112 were correct). In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula to calculate the first 140 digits, of which the first 126 were correct. In 1841, William Rutherford calculated 208 digits, of which the first 152 were correct. The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter (93billion light-years) to a precision of less than one Planck length (at , the shortest unit of length expected to be directly measurable) using expressed to just 62 decimal places. The English amateur mathematician William Shanks, a man of independent means, calculated to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his calculation erroneous; due to the nature