MEASUREMENT OF A CIRCLE 
51 
sides continually doubled, beginning from a square, (h) by 
circumscribing a similar set of regular polygons beginning 
from a square, it being shown that, if the number of the 
sides of these polygons be continually doubled, more than half 
of the portion of the polygon outside the circle will be taken 
away each time, so that we shall ultimately arrive at a circum 
scribed polygon greater than the circle by a space less than 
any assigned area. 
Prop. 3, containing the arithmetical approximation to 7r, is 
the most interesting. The method amounts to calculating 
approximately the perimeter of two regular polygons of 96 
sides, one of which is circumscribed, and the other inscribed, 
to the circle; and the calculation starts from a greater and 
a lesser limit to the value of V 3, which Archimedes assumes 
without remark as known, namely 
265 - ^ 1351 
153 <■ v - 7 g0 • 
How did Archimedes arrive at these particular approxi 
mations? No puzzle has exercised more fascination upon 
writers interested in the history of mathematics. De Lagny, 
Mollweide, Buzengeiger, Hauber, Zeuthen, P. Tannery, Heiler- 
mann, Hultsch, Hunrath, Wertheim, Bobynin: these are the 
names of some of the authors of different conjectures. The 
simplest supposition is certainly that of Hunrath and Hultsch, 
who suggested that the formula used was 
a ± vr > V{a? + b) > a + - 5 
where a 2 is the nearest square number above or below a? + b, 
as the case may be. The use of the first part of this formula 
by Heron, who made a number of such approximations, is 
proved by a passage in his Metrica 1 , where a rule equivalent 
to this is applied to a/720; the second part of the formula is 
used by the Arabian AlkarkhI (eleventh century) who drew 
from Greek sources, and one approximation in Heron may be 
obtained in this way. 2 Another suggestion (that of Tannery 
1 Heron, Metrica, i. 8. 
2 Stereom. ii, p. 184. 19, Hultsch; p. 154. 19, Heib. ^/54 = 7J = 7 T ^ 
instead of 7j\ . 
E 2