Archimedes and the Square Root of 3

One of the most frequently discussed questions in the history of 
mathematics is the "mysterious" approximation of sqrt(3) used by 
Archimedes in his computation of pi.  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 pi] starts from a greater and lesser 
  limit to the value of sqrt(3), which Archimedes assumes 
  without remark as known, namely (265/153) < sqrt(3) < 
  (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^-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 sqrt(3), 
  namely (1351/780) > sqrt(3) > (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  >  sqrt(A) >  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 sqrt(3) 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 PI, 1977


  "Archimedes....takes, in fact, sqrt(3) = 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


Both Boyer and Sondheimer refer to the "Babylonian method" of 
extracting square roots, with Boyer stating that Archimedes' method 
was similar, while Sondheimer suggets 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 sqrt(A), take a_1 as the first 
approximation.  Then iteratively compute 

               a_(n+1) = (a_n + (A/a_n))/2

However, there seems to be some confusion in Boyer's discussion of the 
approximation for sqrt(2) 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:

            1       24       51       10
           ---  +  ----  +  ----  +  ----
            1       60      60^2     60^3

which is written as 1;24,51,10.  Boyer says this value is approximately 
1.414222, which differs from the true sqrt(2) by about 8/10^-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 sqrt(2) by 6/10^-7.  Boyer's decimal value 
1.414222 actually corresponds to 1;24;51;12.  (A new edition of Boyer's 
History has recently come out, but I haven't checked to see if this 
error has been corrected.)

The matter is further confused by Boyer's assertion that the Babylonian 
value for sqrt(2) a_3 from the iteration based on a_1=3/2.  But this 
cannot be true, because all the iterates beginning from 3/2 will be 
slightly ABOVE sqrt(2), whereas 1;24,51,10 is slightly BELOW sqrt(2).  
Also, if you iterate backwards from Boyer's value of a_3=1.414222 you 
deduce that a_1=1.5376918, which does not seem like a natural starting 
point.

Anyway, it seems clear that whatever precise method was used, it was 
related to the continued fraction expansion of sqrt(3), 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 hard for the Greeks to 
perform because of all the long divisions required.  

I suggest that the Greeks may have proceded as follows:  The square 
root of A can be broken into an integer part and a remainder, i.e., 
sqrt(A) = 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

            s(i) = 2N s(i-1) + (A-N^2) s(i-2)

It's easy to see that the value of (A-N^2)(s(i)/s(i+1)) approaches 
r as n goes to infinity.  This is a form of the so-called "ladder 
arithmetic", of which some examples from ancient Babylonia have 
survived.

As an example, to find sqrt(3) we have A=3 and N=1, so the recurrence 
formula is simply s(i) = 2s(i-1) + 2s(i-2).  If we choose the initial 
values s(0)=0 and s(1)=1, the subsequent values in the sequence are

 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920,...

The consecutive terms 18272 and 49920 give r=571/780, which gives 
sqrt(3) = 1+r = 1351/780, Archimedes' upper bound.  Similarly the 
consecutive terms 896 and 2448 gives the lower bound used by 
Archimedes.  (Admittedly, if they used this method, it isn't clear 
why they didn't chosen the lower bound 989/571 based on 6688 and 
18272, unless for some reason they wanted the reduced denominators 
to be divisible by 3.)

The main benefit of this approach is that is relies only on 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

  0  1  2  6
        1  3  8  22
              4  11  30  82
                     15  41 112 306
                             56 153 418  836
                                    209  571  1560  4262

It's too bad Archimedes didn't quote a few more square roots so we 
could check some of these reckless speculations.

For a discussion of how Archimedes used his value of sqrt(3) to
estimate the value of PI, see the note Machin's Merit.