354 THREE GEOMETRICAL PROBLEMS [CH. XVI 



L. van Ceulen devoted no inconsiderable part of his life to 

 the subject. In 1596* he gave the result to 20 places of deci- 

 mals: this was calculated by finding the perimeters of the 

 inscribed and circumscribed regular polygons of 60 x 2" sides, 

 obtained by the repeated use of a theorem of his discovery 

 equivalent to the formula 1 —cos A = 2 sin 5 \A. I possess a 

 finely executed engraving of him of this date, with the result 

 printed round a circle which is below his portrait. He died in 

 1610, and by his directions the result to 35 places of decimals 

 (which was as far as he had calculated it) was engraved on his 

 tombstone-f - in St Peter's Church, Leyden. His posthumous 

 arithmetic]: contains the result to 32 places ; this was obtained 

 by calculating the perimeter of a polygon, the number of whose 

 sides is 2 ra , i.e. 4,611686,018427,387904. Van Ceulen also com- 

 piled a table of the perimeters of various regular polygons. 



Willebrord Snell§, in 1621, obtained from a polygon of 2 30 

 sides an approximation to 34 places of decimals. This is less 

 than the numbers given by van Ceulen, but Snell's method 

 was so superior that he obtained his 34 places by the use of a 

 polygon from which van Ceulen had obtained only 14 (or 

 perhaps 16) places. Similarly, Snell obtained from a hexagon 

 an approximation as correct as that for which Archimedes had 

 required a polygon of 96 sides, while from a polygon of 96 sides 

 he determined the value of w correct to seven decimal places 

 instead of the two places obtained by Archimedes. The reason 

 is that Archimedes, having calculated the lengths of the sides 

 of inscribed and circumscribed regular polygons of n sides, 

 assumed that the length of 1/nth of the perimeter of the circle 

 was intermediate between them; whereas Snell constructed 



* Vanden Circkel, Delf, 1596, fol. 14, p. 1 ; or De Cireulo, Leyden, 1619, p. 3. 



t The inscription is quoted by Prof, de Haan in the Messenger of Mathematics, 

 1874, vol. in, p. 25. 



X De Arithmetische en Geometrische Fondamenten, Leyden, 1615, p. 163 ; or 

 p. 144 of the Latin translation by W. Snell, published at Leyden in 1615 under 

 the title Fundamenta Arithmetica et Geometrica. This was reissued, together 

 with a Latin translation of the Vanden Circkel, in 1619, under the title De 

 Cireulo ; in whioh see pp. 8, 29 — 32, 92. 



§ Cyclometriau, Leyden, 1621, p. 55. 



