Archimedes and the Square Root of 3

One of the most frequently discussed questions in the history of mathematics is the "mysterious" approximation of  used by Archimedes in his computation of p.  Here's a review of what several popular books say on the subject:

It would seem...that [Archimedes] had some (at present unknown) method of extracting the square root of numbers approximately.

W.W Rouse Ball, Short Account of The History of Mathematics, 1908

...the calculation [of p] starts from a greater and lesser limit to the value of , which Archimedes assumes without remark as known, namely (265/153) <  < (1351/780).  How did Archimedes arrive at this particular approximation?  No puzzle has exercised more fascination upon writers interested in the history of mathematics...  The simplest supposition is certainly [see Kline below].  Another suggestion...is that the successive solutions in integers of the equations x2-3y2=1 and x2-3y2=-2 may have been found...in a similar way to...the Pythagoreans.  The rest of the suggestions amount for the most part to the use of the method of continued fractions more or less disguised.

T. Heath, A History of Greek Mathematics, 1921

...he also gave methods for approximating to square roots which show that he anticipated the invention by the Hindus of what amount to periodic continued fractions.

E. T. Bell, Men Of Mathematics, 1937

His method for computing square roots was similar to that used by the Babylonians.

C. B. Boyer, A History of Mathematics, 1968

He also obtained an excellent approximation to , namely (1351/780) >  > (265/153), but does not explain how he got this result.  Among the many conjectures in the historical literature concerning its derivation the following is very plausible.  Given a number A, if one writes it as a2 ± b where a2 is the rational square nearest to A, larger or smaller, and b is the remainder, then  a ± b/(2a) >  > a ± b/(2a ± 1).  Several applications of this procedure do produce Archimedes' result.

M. Kline, Mathematical Thought From Ancient To Modern Times, 1972

Archimedes approximated  by the slightly smaller value 265/153...  How he managed to extract his square roots with such accuracy...is one of the puzzles that this extraordinary man has bequeathed to us.

P. Beckmann, A History Of p, 1977

Archimedes....takes, in fact,  = 1351/780, a very close estimate...but does not say how he got this result, and there has been much speculation on this question.

Sondheimer and Rogerson, Numbers and Infinity, 1981

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