CH. XVI] THREE GEOMETRICAL PROBLEMS 355 



from the sides of these polygons two other lines which gave 

 closer limits for the corresponding arc. His method depends 

 on the theorem 3 sin 0/(2 + cos 0) < < (2 sin §0 + tan $ 0), by 

 the aid of which a polygon of n sides gives a value of ir correct 

 to at least the integral part of (4 log n - 2305) places of decimals, 

 which is more than twice the number given by the older rule. 

 Snell's proof of his theorem is incorrect, though the result is true. 



Snell also added a table* of the perimeters of all regular 

 inscribed and circumscribed polygons, the number of whose 

 sides is 10 x 2" where n is not greater than 19 and not less than 

 3. Most of these were quoted from van Ceulen, but some were 

 recalculated. This list has proved useful in refuting circle- 

 squares. A similar list was given by James Gregory +. 



In 1630 Grienbergerj, by the aid of Snell's theorem, carried 

 the approximation to 39 places of decimals. He was the last 

 mathematician who adopted the classical method of finding the 

 perimeters of inscribed and circumscribed polygons. Closer 

 approximations serve no useful purpose. Proofs of the theorems 

 used by Snell and other calculators in applying this method 

 were given by Huygens in a work§ which may be taken as 

 closing the history of this method. 



In 1656 Wallis|| proved that 



tt_ 2.2.4.4.6.6... 

 2~1.3.3.5.5.7.7...' 



and quoted a proposition given a few years earlier by Viscount 

 Brouncker to the effect that 



4 V 3 a 5j> 



w + 2 + 2 + 2 + ..., 



* It is quoted by Montuola, ed. 1831, p. 70. 



+ Vera Circuit et Hyperbolae Quadratura, prop. 29, quoted by Huygens, 

 Opera Varia, Leyden, 1724, p. 447. 



J Elementa Trigonometrica, Rome, 1630, end of preface. 



§ De Circula Magnitudine Inventa, 1654; Opera Varia, pp. 351 — 387. The 

 proofs are given in G. Pirie's Geometrical Methods of Approximating to the Value 

 oft, London, 1877, pp. 21—23. 



|| Arithmetica Infinitorum, Oxford, 1656, prop. 191. An analysis of the 

 investigation by Wallis was given by Cayley, Quarterly Journal of Mathematics, 

 1889, vol. xxm, pp. 165—169. 



