Kevin Brown documents the difficulty in solving Archimedes square root method related to improving estimates for pi
http://www.mathpages.com/home/kmath038/kmath038.htm
Pat's blog http://pballew.blogspot.com/2010/02/archimedes-and-calculus.html stresses the infinite series side of Archimedes question, without discussing Archimedes finite solution:
4A/3 + A + A/4 + A/12
Please follow the equation of our method, which has a precise sqrt, in the limits of pi by Archimedes:
sqrt(96+1/2+1/96+57933225/96x10000000000000000x3)=1,7724563640625
Geometrical applications: http://squaringofthecircle.gr/?page_id=241&lang=en
Theory: http://squaringofthecircle.gr/?page_id=99&lang=en
Unresolved aspects of ancient ESTIMATION OF PI problem were reported by Kevin Brown and E.B. Davis with upper and lower limits;
(1351/780)^2 is greater than PI is greater than (265/153)^2
A. Archimedes calculated the higher PI limit(1351/780)^2 by:
1. step 1. guess (1 + 2/3)^2 = 1 + 4/3 + 4/9 = 2 + 3/9 + 4/9, meant 2/9 = error1
2. step 2 reduced error1 2/9
by dividing 2/9 by 2(1 + 2/3)
steps that meant
2/9 x (3/10) = 1/15
such that
(1 + 2/3 + 1/15)^2, error2 (1/15)^2 = 1/225 = error2
knowing (1 + 11/15) = 26/15
3. step 3 reduced error2 = 1/225 by dividing by 2 x (26/15) = 52/15
1/225 x (15/52) = 1/15 x (1/52) = (1/780)^2 = error 3
reached
(26/15 - 1/780)^2 = (1351/780)^2 in modern fractions
recorded a unit fraction series that began with step 2 data and subtracted 1/780
(1 + 2/3 + 1/15 - 1/780)^2
as Archimedes would have written
(1 + 2/3 + 1/30 + (13 + 6 + 4 + 2)/780)^2 = (1 + 2/3 + 1/30 + 1/60 + 1/+ 1/195 + 1/390)^2
B . The lower limit 265/153 modified step 2, used
1/17 rather than 1/15, (1+ 2/3 + 1/17) = (1 + 37/51)
such that (1 + 111/153)changed to (1 + 112/153) = 265/153
Archimedes square root was used to improve upper and lower limits of pi ... by beginning with 22/7
Archimedes calculus 4/A/3 = A + A.4 + A/12 offers a different topic
Kevin Brown discussed once unsolved aspects of Archimedes upper and lower limits to pi ... per http://mathpages.com/home/kmath038/kmath038.htm
today the problem has been solved as cited above
footnote: Fibonacci’s square root of 17 method was appropriately cited as used by Galileo though not properly analyzed in every detail. Fibonacci guessed (4 + 1/8)^2 = (17 + 1/64) , and Fibonacci reduced the estimated 1/64 error by finding an inverse proportion:1/64 x 8/66 = 1/528 which meant (4 + 1/8 - 1/528)^2 = (2177/528)^2 = 17.000003 is accurate to (1/528)^2 per Archimedes and not Newton.
III CONCLUSION
Unit fraction square root was formalized by 2050 BCE and used by Egyptians, Greeks, Arabs, medieval scribes and as late as Galileo. The method estimated irrational square roots of N by 1-step, 2-step, 3-step and 4-steps methods. Step 1 guessed quotients (Q) and remainders (R) = n/(2Q) with n = (N - Q^2). Step 2, 3, and 4 reduced error 1, 2 and 3 associated with the previous step by dividing by 2(Q + R).
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