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 x^{2}3y^{2}=1 and x^{2}3y^{2}=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 a^{2} ± b where a^{2} 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 

Kline doesn’t specify the initial estimate, and doesn’t address the proliferation of values produced by his suggested method (doubling the number of upper and lower bounds on each step). Both Boyer and Sondheimer refer to the "Babylonian method" of extracting square roots, with Boyer stating that Archimedes' method was similar, while Sondheimer suggests that, due to the primitive number system used by the Greeks, Archimedes would have had difficulty with the complicated fractions involved in the Babylonian method. 

Both authors describe the "Babylonian method" (also called Newton's method) as follows: To find _{}, take a_{1} as the first approximation. Then iteratively compute 

_{} 

However, there seems to be some confusion in Boyer's discussion of the approximation for _{} used by the Babylonians. The value he cites from the Old Babylonian tablet No. 7289 from the Yale collection is interpreted as the number, expressed in the base 60, shown below: 

_{} 

which is written as 1;24,51,10. Boyer says this value is approximately 1.414222, which differs from the true _{} by about (8.4)10^{6}. The problem is that the sexigesimal value 1;24;51;10 actually corresponds to the decimal 1.4142129 (as correctly stated by Sondheimer), which differs from the true _{} by –(5.99)10^{7}. Boyer's decimal value 1.414222 actually corresponds to 1;24;51;12. The matter is further confused by Boyer's assertion that the Babylonian value for _{} is a_{3} from the iteration based on a_{1} = 3/2, which cannot be true, because all the "a" iterates beginning from 3/2 will be slightly above _{}, whereas 1;24,51,10 is slightly below _{}. The value of a_{3} produced by the Babylonian algorithm is actually 577/408 = 1.4142156, which has the sexigesimal expansion 1:24,51,10,35,17,... On the other hand, if we iterate backwards from Boyer's value of a_{3} = 1.414222 we deduce that a_{1} = 1.5376918, which does not seem like a natural starting point. 

In any case, it seems clear that whatever precise method was used, it was related to the continued fraction expansion of _{}, which of course is closely connected to the Pell equation x^{2}  3y^{2} = 1. (The latter naturally arises if we seek a rational square (x/y)^{2} just slightly greater then 3, which means we want the integer x^{2} to be just slightly greater than the integer 3y^{2}. Setting this difference to 1 gives the Pell equation.) Otherwise it would be very hard to explain how they could have arrived at the two convergents 265/153 and 1351/780, each of which is a "best rational approximation" up to the respective denominators. However, I agree with Sondheimer that an explicit continued fraction algorithm would have been difficult for the Greeks to perform because of all the long divisions required. 

One possible method that could have been used by the Greeks is as follows: The square root of A can be broken into an integer part and a remainder, i.e., _{} = N + r where N is the largest integer such that N^{2} is less than A. The value of r can be approximated to any desired degree of precision using only integer additions and multiplications based on the recurrence formula 

_{} 

It's easy to see that the value of (AN^{2})(s_{n}/s_{n+1}) approaches r as n goes to infinity. This is a form of the socalled "ladder arithmetic", of which some examples from ancient Babylonia have survived. As an example, to find _{} we have A = 3 and N = 1, so the recurrence formula is simply s_{n} = 2s_{n}_{1} + 2s_{n}_{2}. If we choose the initial values s_{0} = 0 and s_{1} = 1, the subsequent values in the sequence are 

_{} 

The consecutive terms 18272 and 49920 give r = 571/780, which gives _{} = 1 + r = 1351/780, Archimedes' upper bound. Similarly the consecutive terms 896 and 2448 gives the lower bound used by Archimedes. The main benefit of this approach is that it relies on only simple integer operations. The size of the integers could have been kept small by eliminating the accumulating powers of 2 at each stage as follows 

_{} 

However, if they used this method, it isn't clear why they didn't choose the lower bound 989/571 based on 6688 and 18272. So, although this method was surely within their capability, it doesn’t seem likely to have been the source of Archimedes’ values. 

To my mind, the most plausible source of Archimedes’ upper and lower bounds is a simple linear fractional iteration. Imagine that their first estimate for the square root of 3 was 5/3, perhaps based on the fact that 5^{2} = 25 is close to 3(3^{2}) = 27. From here it isn't hard to see that if x is a bound on the square root of 3, then (5x+9)/(3x+5) is a closer bound on the opposite side. Letting e denote the error x^{2}  3 for the estimate x, the error of the next estimate is 

_{} 

Thus the error is negated and reduced by a factor of nearly 52 on each step. Beginning with x = 5/3, the sequence of iterates of x → (5x+9)/(3x+5) is 

_{} 

which gives the Archimedean lower and upper bounds as the 2nd and 3rd iterates. Given their limited facility for numerical calculation, it's easy to understand why they wouldn't have gone on to compute 13775/7953 or any higher iterates. 

For a discussion of how Archimedes used his value of _{} to estimate the value of PI, see the note Machin's Merit. 
