A history of Greek mathematics - Wilbourhall.org

sides continually doubled, beginning from a square, (b) by
circumscribing a similar set of regular polygons beginning
from a square, it being shown that, if the number of the
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
1
MEASUREMENT OF A CIRCLE 51
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 circumscribed
polygon greater than the circle by a space less than
any assigned area.
Prop. 3, containing the arithmetical approximation to n, 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,
265 s- a/Q ^ 1351
IT'S a
v
2a -* — ±
' 2a+
mations? No puzzle has exercised more fascination upon
writers interested in the history of mathematics. De Lagny,
simplest supposition is certainly that of Hunrath and Hultsch,
who suggested that the formula used was
where a 2 is the nearest square number above or below a 2 ± 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 \/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 = 7^ = 7^
instead of 7 I
5
1
.
E 2