THE SQUARING OF THE CIRCLE. 



I2 9 



passed even those of the Greeks. In the most ancient source of 

 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 prob- 

 lem is dealt with, which might fittingly be termed the circling of 

 the square. The half of the side of a given square is prolonged in 

 length one third of the excess 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 n 

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

 find out in this way that the value of n upon which the Indian cir- 

 cling of the square is based, is about from five to six hundredths 

 smaller than the true value, whereas the approximate n of Archi- 

 medes, 3^, is only from one to two thousandths too large, and that 

 the old Egyptian value exceeds the true value by from one to two 

 hundredths. 



Cyclometry very probably 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 approxima- 

 tion that in exactness surpasses even that of Ptolemy. The Hindu 

 result gives 3-1416 for TT, while n really lies between 3-141592 and 

 3-141593. How the Hindus obtained this excellent value is told by 

 Gane9a, the commentator of Bhaskara, an author of the twelfth 

 century. Gane9a says that the method of Archimedes was carried 

 still farther by the Hindu mathematicians ; that by continually 

 doubling the number of sides they proceeded from the hexagon to 

 a polygon of 384 sides, and that by the comparison of the circum- 

 ferences of the inscribed and circumscribed 384-sided polygons they 

 found that n was equal to 3927:1250. It will be seen that the value 

 given by Bhaskara is identical with the value of Aryabhatta. It is 

 further worthy of remark that the earlier of these two Hindu math- 

 ematicians does not mention either the value 3^ of Archimedes or 

 the value 3^ of Ptolemy, but that the later one knows of both 

 values and especially recommends that of Archimedes as the most 



