THE SQUARING OF THE CIRCLE. 113 



the comparison of lunes. (rregory of St. Vinceut published a quadrature 

 the error of which was very hard to detect, but was finally discovered 

 by Descartes. 



Peter 3Ietius, and Vieta. — Of the famous mathematiciaus who dealt 

 with our problem in the period between the close of the fifteenth cen- 

 tury 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 only small numbers. The problem 

 received a different advancement at the hands of the famous mathema- 

 tician Vieta. Vieta was the first to whom the idea occurred of repre- 

 senting 7t with mathematical exactness by an infinite series of coutin- 

 uable operations. By comparison of inscribed and circumscribed 

 polygons, Vieta found that we approach nearer antl nearer to n if we 

 allow the operations of the extraction of the square root of h and of 

 addition and of multiplication to succeed each other in a certain man- 

 ner, and that - must come out exactly if this series of operations could 

 be indefinitely continued. Vieta thus found that to a diameter of 10,(KK) 

 million units a circumference belongs of 31,415 million, and from 

 926,535 to 926,536 units of the same length. 



Adrianus Romanus, Ludolf Van Ceulen. — But Vieta was outdone by 

 the Netherlander Adrianus Romanus, who added fiveadditional decimal 

 places to the ten of Vieta. To accomplish this he computed with un- 

 speakable labor the circumference of a regular circumscribed polygon 

 of 1,073,741,824 sides. This number is the thirtieth power of 2. Yet 

 great as the labor of Adrian us Eomanus was, that of Ludolf Van Ceu- 

 len was still greater, for the latter calculator succeeded in carrying the 

 Archimedean process of approximation for the value of ti to 35 decimal 

 places, that is, the deviation from the true value was smaller than one 

 one thousand quintillionth, a degree of exactness that we can hardly 

 have any conception of. Ludolf published the figures of the tremendous 

 computation that led to this result. His calculation was carefully ex- 

 amined by the mathematician Griemberger and declared to be correct. 

 Ludolf was justly proud of his work, and, following the example of 

 Archimedes, requested in his will that the result of his most important 

 mathematical performance, the computation of ;rto 35 decimal places, 

 be engraved upon his tombstone, a request which is said to iiave been 

 carried out. In honor of Ludolf, tt is called today in Germany the 

 Ludolfiau number. 



The new method of Snell. Huygens\s verification of it. — Although 

 through the labor of Ludolf a degree of exactness for cyclometrical 

 operations was now obtained that was more than sufficient for any 

 practical purpose that could ever arise, neither the problem of construc- 

 tive rectification nor that of constructive quadrature was thereby in 

 any respect theoretically advanced. The investigations conducted by 

 the famous mathematicians and i>hysicists Uuygens and Snell, about 

 the middle of the seventeenth century, were more important from amathe- 

 H. Mis. 129 8 



