110 THE Sl^lARING OF THE CIRCLE. 



inedes was carried still farther by the Hindu mathematicians; that by 

 continually doubling: the number of sides they proceeded from the hex- 

 agon to a polyuon of .'i.S4 sides, and that by the comparison of the cir- 

 cumferences of the inscribed and circumscribed 384 sided polygons they 

 found that n was ecpial to 31»27 — 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 mathe- 

 maticians does not mention either the value Sj of Archimedes or the 

 value 3jyg of Ptolemy, but that the later knows of both values and 

 especially recommends that of Archimedes as the most useful one for 

 practical application. Strange to say, the good approximate value of 

 Arybhatta does not occur in Branmgupta, the great Hindu mathema- 

 tician who flourished in the beginning of the seventh century; but we 

 find the curious information in this author that the area of a circle is 

 exactly equal to the square root of 10 when the radius is unity. The 

 value of n as derivable from this formula (a value from two to three 

 hundredths too large) has unquestionably arisen upon Hindu soil, for 

 it occurs in no Grecian mathematician; and Arabian authors, who were 

 in a better position than we to know Greek and Hindu mathematical 

 literature, declare that the approximation which uiakes n equal to the 

 square root of 10 is of Hindu origin. It is possible that the Hindu 

 people, who were addicted more than any other to numeral mysticism, 

 sought to fiud in this ai)proximation some connection with the fact that 

 man has ten fingers ; and ten accordingly is the basis of their numeral 

 system. 



Keviewing the achievements of the Hindus generally with respect to 

 the problem of the quadrature, we are brought to recognize that this 

 ])eople, whose talents lay more in the line of arithmetical computation 

 than in the perception of spatial relations, accomplished as good as noth- 

 ing on the pure geometrical side of the problem, but that the merit be- 

 longs to them of having carried the Archimedean method of computing n 

 several stages farther, and of having obtained in this way a much more 

 exact value for it; — a circumstance that is explainable when we consider 

 that the Hindus are the inventors of our present system of numeral 

 notation, possessing which they easily outdid Archimedes, who em- 

 ployed the awkward Greek system. 



Among the Chinese. — With regard to the Chinese, this people oi)erated 

 in ancient times with the Babylonian value for r, or 3, but possessed 

 knowledge of the approximate value of Archimedes, at least since the 

 end of the sixth century. Besides this, tiiere appears in a number of 

 Chinese mathemaHcal treatises an approximate value peculiarly their 

 own, in wlii(;h -=3^'j, ; a value, however, which, notwithstanding it is 

 written in large figures, is no better than that of Archimedes. At- 

 tempts at the constructive quadrature of the circle are not found among 

 the Chinese. 



