132 THE SQUARING OF THE CIRCLE. 



believed the famous Cardinal, and marvelled at his wisdom, until 

 Regiomontanus, in letters written in 1464 and 1465 and published 

 m J 533> rigorously demonstrated that the Cardinal's quadrature 

 was incorrect. The construction of Cusa was as follows. The ra- 

 dius of a circle is prolonged a distance equal to the side of the in- 

 scribed square; the line so obtained is taken as the diameter of a 

 second circle, and in the latter an equilateral triangle is described; 

 then the perimeter of the latter is equal to the circumference of the 

 original circle. If this construction, which its inventor regarded as 

 exact, be considered as a. construction of approximation, it will be 

 found to be more inexact even than the construction resulting from 

 the value 7t = $\. For by Cusa's method n would be from five to 

 six thousandths smaller than it really is. 



In the beginning of the sixteenth century a certain Bovillius 

 appears, who also gave the construction of Cusa, this time with- 

 out notice. But about the middle of the sixteenth century a book 

 was published which the scholars of the time at first received with 

 interest. It bore the proud title De Rebus Mathematicis Hactenus 

 Desideratis. Its author, Orontius Finaeus, represented that he had 

 overcome all the difficulties that had ever stood in the way of geo- 

 metrical investigators ; and incidentally he also communicated to 

 the world the "true quadrature" of the circle. His fame was short- 

 lived. For soon afterwards, in a book entitled De Erratis Orontii, 

 the Portuguese Petrus Nonius demonstrated that Orontius's quad- 

 rature, like most of his other professed discoveries, was incorrect. 



In the succeeding period the number of circle-squarers so in- 

 creased that we shall have to limit ourselves to those whom mathe- 

 maticians recognise. And particularly is Simon Van Eyck to be 

 mentioned, who towards the close of the sixteenth century pub- 

 lished a quadrature which was so approximate that the value of 7t 

 derived from it was more exact even than that of Archimedes ; and 

 to disprove it the mathematician Peter Metius was obliged to seek 

 a still more accurate value than 3^. The erroneous quadrature of 

 Van Eyck was thus the occasion of Metius's discovery that the ra- 

 tio 355:113, or 3^%' var i e< i from the true value of it by less than 

 one one-millionth, eclipsing accordingly all values hitherto ob- 



