Legcndre's Geometry. 297 



tion, though it has exhausted all the resources of human 

 skill and invention, and it strongly reminds us, that imperfec- 

 tion is attached even to the most certain and most perfect 

 of the sciences. The approximation of this ratio, has, 

 however, been carried so far, that if it were exactly known, 

 it would have no practical advantage over the approximate 

 ratio. It would now be considered absurd, to spend much 

 time in attempting to square the circle. 



Archimedes obtained the ratio of \^, which is sufficient- 

 ly near for common purposes, and has been much used. 

 Metius gave a much more exact value of this ratio in the 

 expression ^ff. Other mathematicians have found the 

 value of the circumference, when the diameter is unity, 

 3,141592653 Sjc. Euler gives an approximation of this ra- 

 tio which extends to 127 decimal places,* and this number 

 has been extended even to 1 40 places. The roots ofimperfect 

 powers are not known with greater exactness, than this ratio. 



Legendre has not given the ratio, that has lately been 

 discovered by the English in their researches into the learn- 

 ing of the Eastern Indians, but we think it ought to have a 

 place, both on account of its exactness and its remarkable 

 origin. This ratio, which is yf f|, is contained in a work of 

 the Brachmans entitled Ayeen Akbery, and is not only much 

 more approximate, but also is regarded by them as more 

 ancient than that of Archimedes. It is, doubtless, to be re- 

 garded as a part of the immense wreck of ancient learning 

 which is scattered all over India. In that interesting coun- 

 try, "we every where find methods of calculation without 

 the principles on which they are founded ; rules blindly 

 followed without being understood ; phenomena without 

 their explanation; and elements carefully determined, 

 while others more important, and equally obvious, are alto- 

 gether unknown. "f The Indian ratio corresponds to 3.1416, 

 and must have depended on a polygon of 768 sides, where- 

 as that of Archimedes depends upon one of 96 sides. 



Two lemmas are given as the basis of the investigation 

 of the measure of the circle, which is otherwise conducted 

 after the manner of Archimedes. Two methods of approx- 

 imation are there given for its quadrature. An appendix 



* Introduction a I'analyse infinitesimale, Tome I. p. 92, 



tEdinb. Encyc, Vol. II. p. 550, 

 Vol. VI.— -No. 2. 38 



