Date: Jan 24, 2013 10:21 AM
Author: Milo Gardner
Subject: Archiimedes pi estimated range

Archimedes estimated pi by

22/7 > pi > 223/71

1. the method scaled 22/7 by 10/10 = 220/70

2.obtained the lower limit by 22/7(10/10) = 220/70

220/71 and added 223/71

as the root of 3 was estimated by

Archimedes' square root of 3 cited limits

(1271/780)^2 > 3 > (265/153)^2

A. The higher limit 1251/780 was calculated in three steps

1. (1 + 2/3)^2 error 1 = 2/9

2. 2/9 x (3/10) = 1/15, (1 + 2/3 + 1/15)= (1 + 11/15). error 2 = 1/225

3. 1/225 x (15/52) = 1/780, (1 + 11/15 -1/780) = 1251/780

B. the lower limit 265/153 modified step 2, used

2. 1/17 rather than 1/15, (1+ 2/3 + 1/17) = (1 + 37/51)

such that (1 + 111/153) increased to (1 + 112/153) = 265/153

Q.E.D.