134 THE SQUARING OF THE CIRCLE. 



in a certain manner, and that n must come out exactly, if this series 

 of operations could be continued indefinitely. Vieta thus found 

 that to a diameter of 10000 million units a circumference belongs of 

 from 31415 million 926535 units to 31415 million 926536 units of 

 the same length. 



But Vieta was outdone by the Netherlander Adrianus Romanus, 

 who added five additional decimal places to the ten of Vieta. To 

 accomplish this he computed with unspeakable labor the circum- 

 ference of a regular circumscribed polygon of 1073741824 sides. 

 This number is the thirtieth power of 2. Yet great as the labor of 

 Adrianus Romanus was, that of Ludolf Van Ceulen was still greater; 

 for the latter calculator succeeded in carrying the Archimedean 

 process of approximation for the value of n 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 have 

 scarcely any conception of. Ludolf published the figures of the 

 tremendous computation that led to his result. His calculation 

 was carefully examined by the mathematician Griemberger and de- 

 clared to be correct. Ludolf was justly proud of his work, and fol- 

 lowing the example of Archimedes, requested in his will that the 

 result of his most important mathematical performance, the com- 

 putation of 7t to 35 decimal places, be engraved upon his tomb- 

 stone ; a request which is said to have been carried out. In honor 

 of Ludolf, n is called to-day in Germany the Ludolfian number. 



Although through the labor of Ludolf a degree of exactness for 

 cyclometrical operations was now obtained that was more than suf- 

 ficient for any practical purpose that could ever arise, neither the 

 problem of constructive rectification nor that of constructive quad- 

 rature had been in any respect theoretically advanced thereby. The 

 investigations conducted by the famous mathematicians and phys- 

 icists Huygens and Snell about the middle of the seventeenth cen- 

 tury, were more important from a mathematical point of view than 

 the work of Ludolf. In his book Cyclometricus Snell took the posi- 

 tion that the method of comparison of polygons, which originated 

 with Archimedes and was employed by Ludolf, was not necessarily 

 the best method of attaining the end sought ; and he succeeded by 



