THE SQUARING OF THE CIRCLE. 13! 



Archimedes. Attempts at the constructive quadrature of the circle 

 are not found among the Chinese. 



Greater were the merits of the Arabians in the advancement of 

 mathematics; and especially in virtue of the fact that they pre- 

 served from oblivion the results of both Greek and Hindu research 

 and handed them down to the Christian countries of the West. The 

 Arabians expressly distinguished between the Archimedian approx- 

 imate value and the two Hindu values, the square root of 10 and 

 the ratio 62832:20000. This distinction occurs also in Muhammed 

 Ibn Musa Alchwarizmi, the same scholar who in the beginning of 

 the ninth century brought the principles of our present system of 

 numerical notation from India and introduced it into the Moham- 

 medan world. The Arabians, however, did not study the numerical 

 quadrature of the circle only, but also the constructive ; for in- 

 stance, an attempt of this kind was made by Ibn Alhaitam, who 

 lived in Egypt about the year 1000 and whose treatise upon the 

 squaring of the circle is preserved in a Vatican codex, which un- 

 fortunately has not yet been edited. 



Christian civilisation, to which we are now about to pass, pro- 

 duced up to the second half of the fifteenth century extremely in 

 significant results in mathematics. Even with regard to our pres- 

 ent problem we have but a single important work to mention ; the 

 work, namely, of Frankos von Liittich on the squaring of the circle, 

 published in six books, but preserved only in fragments. The 

 author, who lived in the first half of the eleventh century, was 

 probably a pupil of Pope Sylvester II., who was himself a not in- 

 considerable mathematician for his time and the author of the most 

 celebrated geometrical treatise of the period. 



Greater interest came to be bestowed upon mathematics, and 

 especially on the problem of the quadrature of the circle, in the 

 second half of the fifteenth century, when the sciences again began 

 to revive. This interest was principally aroused by Cardinal Nico- 

 las de Cusa, a man highly esteemed for his astronomical and calen- 

 darial studies. He claimed to have discovered the quadrature of 

 the circle by employing only straight edge and compasses and thus 

 attracted the attention of scholars to the historic problem. People 



