. Legendre’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 22, 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 323, Other mathematicians have found the 
value of the circumference, when the diameter is unity, 
3,141592653 &c. Euler gives an approximation of this ra- 
tio which extends to 127 decimal places,* and this number 
has been extendedevento 140places. 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 $227, is contained ina work of 
the Brachmans entitled 4yeen 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.”+ 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 4 analyse infinitésimale, Tome I. p. 92: 
+ Edinb. Encyc. Vol. lf. p. 550. 
Vo. VI.— No. 2. 38 
