THE SQUARING OF THE CIRCLE. 109 



a matter of fact, the expression 3 + -g-o + 3 too = ^tto represents the 

 uuniber - more exactly than 3|; but on the other hand is, by reason of 

 the magnitude of the numbers 17 and 120 as compared with the num- 

 bers 1 and 7, more cumbersome. 



Among the Romans. — In the mathematical sciences, more than in any 

 other, the Komans stood upon the shoulders of the Greeks. Indeed, with 

 respect to cyclometry,they not only did not add anything to the Grecian 

 discoveries, but often evinced even that they either did not know of the 

 beautiful result obtained by Archimedes or at least did not know how to 

 appreciate it. For instance, Vitruvius, who lived during the time ot 

 Augustus, computed that a wheel 4 feet in diameter must measure 12i 

 feet in circumference; in other words, he made 7t equal to 3^. And, 

 similarly, a treatise on surveying, preserved to us in the Gudian manu- 

 script of the library at Woifenbiittel, contains the following instructions 

 to square the circle : Divide the circumference of a circle into four parts 

 and make one part the side of a square ; this square will be equal in 

 area to the circle. Aside from the fact that the rectification of the arc 

 of a circle is requisite to the construction of a square of this kind, the 

 Roman quadrature, viewed as a calculation, is more inexact even than 

 any other computation ; for its result is that tt z= i. 



AmoHfi the Hindus. — The mathematical performances of the Hindus 

 were not only greater than those of the Eomans, but in certain directions 

 even surpassed those of the Greeks. In the most ancient source for the 

 mathematics of India that we know of, the Culvasutras, which date back 

 to a little before our chronological era, we do not find, it is true, the 

 squaring of the circle treated of, but the opposite problem is dealt v^ith 

 which might fittingly be termed the circling of the square. The half of 

 the side of a given square is prolonged one-third of the excess in length 

 of half the diagonal over half the side, and the line thus obtained is taken 

 as the radius of the circle equal in area to the square. The simplest way 

 to obtain an idea of the exactness of this construction is to compute how 

 great rr would have to be if the construction were exactly correct. We 

 find out in this way that the value of tt, upon which the Indian circling of 

 the square is based, is about from five to six hundredths smaller than the 

 true value, whereas the approximate tt of Archimedes, 3\, is only from 

 one to two thousandths too large, and the old Egyptian value exceeds 

 the true value by from one to two hundredths. Cyclometry very prob- 

 ably made great advances among the Hindus in the first four or five 

 centuries of our era; for Aryabhatta, who lived about the year 500 after 

 Christ, states that the ratio of the circumference to the diameter is 

 62832 -^ 20000, an approximation that in exactness surpasses even that of 

 Ptolemy. The Hindu result gives 34416 for tt, while 7t really lies be- 

 tween 3-141592 and 3-141593. How the Hindus obtained this excellent 

 approximate value is told by Gane9a, the commentator of Bhaskara, an 

 author of the twelfth century. Ganega says that the method of Archi- 



