CH. XVI] THREE GEOMETRICAL PROBLEMS 353 



Leonardo of Pisa*, in the thirteenth century, gave for v the 

 value 1440/458^, which is equal to 31418.... In the fifteenth 

 century, Purbachf gave or quoted the value 62832/20000, 

 which is equal to 31416; Cusa believed that the accurate 

 value was f (V3 + V6), which is equal to 31423...; and, in 

 1464, RegiomontanusJ is said to have given a value equal to 

 314243. 



Vieta§, in 1579, showed that ir was greater than 

 31415926535/10 10 , and less than 31415926537/10 10 . This was 

 deduced from the perimeters of the inscribed and circumscribed 

 polygons of 6 x 2 16 sides, obtained by repeated use of the formula 

 2 sin 2 \6 = 1 - cos 6. He also gave|| a result equivalent to the 

 formula 



2 _ J2 V(2 + V2) V(2 + V(2 + V2)} 

 ir 2 2 2 



The father of Adrian MetiusIT, in 1585, gave 355/113, 

 which is equal to 314159292..., and is correct to six places 

 of decimals. This was a curious and lucky guess, for all that 

 he proved was that ir was intermediate between 377/120 and 

 333/106, whereon he jumped to the conclusion that he would 

 obtain the true fractional value by taking the mean of the 

 numerators and the mean of the denominators of these fractions. 

 In 1593 Adrian Romanus** calculated the perimeter of the 

 inscribed regular polygon of 1073,741824 (i.e. 2 s ") sides, from 

 which he determined the value of ir correct to 15 places of 

 decimals. 



* Boucompagni's Scritti di Leonardo, vol. ii (Practica Qeometriae), Rome, 

 1862, p. 90. 



t Appendix to the De Triangulis of Regiomontanus, Basle, 1541, p. 131. 



X In his correspondence with Cardinal Nicholas de Cusa, Be Quadratura 

 Circuit, Nuremberg, 1533, wherein he proved that the cardinal's result was 

 wrong. I cannot quote the exact reference, but the figures are given by com- 

 petent writers and I have no doubt are correct. 



§ Canon Mathematical sen ad Triangula, Paris, 1579, pp. 56, 66 : probably 

 this work was printed for private circulation only, it is very rare. 



|| Vietae Opera, ed. Schooten, Leyden, 1646, p. 400. 



t Arithmeticae libri duo et Geometriae, by A. Metius, Leyden, 1626, pp. 88— 

 89. [Probably issued originally in 1611.] 



** Ideae Mathematieae, Antwerp, 1593: a rare work, which I have never been 

 able to consult. 



B. R. 23 



