Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/41

Adrianus Romanus (Adriaen van Rooman, born in Lyons, 1561—1615) by the help of a 15 . 2²⁴agon calculated $\pi$ to 15 places of decimals.

Ludolf van Ceulen (Cologne) (1539—1610), after whom the number is still called in Germany "Ludolph's number," is said to have calculated $\pi$ to 35 decimals. According to his wish the value was engraved on his tombstone which has been lost. In his writing Van den Cirkel (Delft, 1596) he explained how, by employing the method of Archimedes, using in- and circum-scribed polygons up to the 60 . 2²⁹agon, he obtained $\pi$ to 20 decimal places. Later, in his work De Arithmetische en Geometrische fondamenten he obtained the limits given by

3 14159265358979323846264338327950/100000000000000000000000000000000

and the same expression with 1 instead of 0 in the last place of the numerator.

The work of Snellius and Huyghens

In a work Cyclometricus, published in 1621, Willebrod Snellius (1580—1626) shewed how narrower limits can be determined, without increasing the number of sides of the polygons, than in the method of Archimedes. The two theorems, equivalent to the approximations

${\frac {1}{3}}(2\sin {\theta }+\tan {\theta })<\theta <3/(2\operatorname {cosec} {\theta }+\cot {\theta })$,

by which he attained this result were not strictly proved by him, and were afterwards established by Huyghens; the approximate formula $\theta ={\frac {3\sin {\theta }}{2+\cos {\theta }}}$ had been already obtained by Nicholas of Cusa (1401—1464). Using in- and circum-scribed hexagons the limits 3 and 3.464 are obtained by the method of Archimedes, but Snellius obtained from the hexagons the limits 3.14022 and 3.14160, closer than those obtained by Archimedes from the 96agon. With the 96agon he found the limits 3.1415926272 and 3.1415928320. Finally he verified Ludolf's determination with a great saving of labour, obtaining 34 places with the 230agon, by which Ludolf had only obtained 14 places. Grunberger^[1] calculated 39 places by the help of the formulae of Snellius.

The extreme limit of what can be obtained on the geometrical lines laid down by Archimedes was reached in the work of Christian Huyghens (1629—1665). In his work^[2] De circuli magnitudine inventa,

1. Elementa Trigonometriae, Rome, 1630.
2. A study of the German Translation by Budio will repay the trouble.