THE SQUARING OF 1HE CIRCLE. 



133 



tained. Moreover, it is demonstrable by the theory of continued 

 fractions, that, admitting figures to four places only, no two num- 

 bers more exactly represent the value of it than 355 and 113. 



In the same way the quadrature of the great philologist Joseph 

 Scaliger led to refutations. Like most circle-squarers who believe 

 in their discovery, Scaliger also was little versed in the elements of 

 geometry. He solved the famous problem, however, at least in 

 his own opinion, -and published in 1592 a book upon it, which 

 bore the pretentious title Nova Cyclometria, and in which the name 

 of Archimedes was derided. The baselessness of his supposed dis- 

 covery was demonstrated to him by the greatest mathematicians of 

 his time ; namely, Vieta, Adrianus Romanus, and Clavius. 



Of the erring circle-squarers that flourished before the middle 

 of the seventeenth century three others deserve particular mention, 

 Longomontanus of Copenhagen, who rendered such great ser- 

 vices to astronomy, the Neapolitan John Porta, and Gregory of St. 

 Vincent. Longomontanus made "'= r 3iVbWV anc ^ was so convinced 

 of the correctness of his result as to thank God fervently, in the 

 preface to his work Inventio Quadraturae Circuit, that He had 

 granted him in his old age the strength to conquer the celebrated 

 difficulty. John Porta followed the example of Hippocrates and 

 endeavored to solve the problem by a comparison of lunes. Gregory 

 of St. Vincent published a quadrature, the error of which was very 

 hard to detect but was finally discovered by Descartes. 



Of the famous mathematicians who dealt with our problem in 

 the period between the close of the fifteenth century and the time 

 of Newton, we first meet with Peter Metius, before mentioned, who 

 succeeded in finding in the fraction 355:113 the best approximate 

 value for n involving small numbers only. The problem received 

 a different advancement at the hands of the famous mathematician 

 Vieta. Vieta was the first to whom the idea occurred of represent- 

 ing n with mathematical exactness by an infinite series of definitely 

 prescribed operations. By comparing inscribed and circumscribed 

 polygons, Vieta found that we approach nearer and nearer to 7t if 

 we cause .the operations of extracting the square root of ^, and 

 certain related additions and multiplications, to succeed each other 



