THE SQUARING OF THE CIRCLE. Ill 



Among the Arabs. — Greater were the merits of the Arabians in the 

 advancement and development of mathematics, and especially in virtue 

 of the fact that they preserved from oblivion both Greek and Himlii 

 mathematics, and handed them down to the Christian countries of the 

 West. The Arabians expressly distinguished between the Archimedean 

 approximate value and the two Hindn values, the square root of 10 and 

 the ratio 62832: 20000. This distinction occurs also in Muhnmraed Ibn 

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

 century brought the principles of our present system of numerical nota- 

 tion from India and introduced the same into the Mohammedan world. 

 The Arabians however studied not only the numerical quadrature of 

 the circle, but also the constructive; as, for instance, 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 has un- 

 fortunately not yet been edited. 



In Christian times. — Christian civilization, to which we are now about 

 to pass, produced up to the second half of the fifteenth century extremely 

 insignificant results in mathematics. Even with regard to our present 

 problem ^e have but a single important work to mention — the work, 

 namely, of Frankos Von Liittich upon the squaring of the circle, pub- 

 lished in six books, but only preserved in fragments. The author, who 

 lived in the first half of the eleventh century, was probably a pupil of 

 Pope Sylvester ii, himself a not inconsiderable mathematician for his 

 time, and who also wrote the most celebrated book on geometry of the 

 period. 



Cardinal Nicolaus de Cusa. — Greater interest came to be bestowed 

 upon mathematics in general, but 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 especially 

 aroused by Cardinal Nicolaus De Cusa, a man highly esteemed on ac- 

 count ot his astronomical and calendarial studies. He claimed to have 

 discovered the quadrature of the circle by the employment solely of 

 compasses and ruler, and thus attracted the attention of scholars to 

 the now historic problem. People believed the famous cardinal and 

 marvelled at his wisdom, until Eegiomontanus, in letters which he wrote 

 in 1464 and 1465, and which were published in 1533, rigidly demon- 

 strated that the cardinal's quadrature was incorrect. The construction 

 of Cusa was as follows:. The radius of a circle is prolonged a distance 

 equal to the side of the inscribed square ; the line thus obtained is taken 

 as the diameter of a second circle, and in the latter an equilateral trian- 

 gle is described ; then the perimeter of the latter is equal to the circum- 

 ference 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 ;: = 3|. For by Cusa's method tt would be from five to 

 six thousandths smaller than it really is. 



