In his famous estimation of $\pi$ by inscribed and circumscribed polygons, Archimedes uses several rational approximations of irrational values; a typical example is that he states, without explanation, $$\sqrt 3 > \frac{265}{153}.$$

The sudden appearance of $\frac{265}{153}$ has puzzled many people through the ages. The page [Archimedes and the Square Root of 3](http://mathpages.com/home/kmath038/kmath038.htm) lists several such. This example, attributed to W.W. Rouse Ball, is typical:

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> It would seem...that [Archimedes] had some (at present unknown) method of extracting the square root of numbers approximately.
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To my mind, there is a very easy answer to this seeming puzzle. If you want rational approximations to $\sqrt 3$, the *very simplest* thing you can possibly do is as follows: You want to find integers $a$ and $b$ whose ratio is close to $\sqrt 3$, or equivalently you want to find $a^2$ and $b^2$ whose ratio is close to $3$. So you should tabulate $n^2$ and $3n^2$ and look for integers in the left column that are approximately equal to integers in the right column:

$$\begin{array}{rr}
n^2 & 3n^2 \\ \hline
1 & \color{darkgreen}{3} \\
\color{darkgreen}{4} & 12 \\
9 & \color{darkred}{27} \\
16 & \color{darkblue}{48} \\
\color{darkred}{25} & 75 \\
36 & 108 \\
\color{darkblue}{49} & 147 \\
64 & 192 \\
81 & 243 \\
100 & 300 \\
121 & \color{purple}{363} \\
\vdots & \vdots \\
\color{purple}{361} & 1183 \\
\vdots & \vdots \\
\end{array}
$$

The pairs of colored entries give the approximations $\color{green}{\frac21}, \color{maroon}{\frac53}, \color{darkblue}{\frac74}, $ and $\color{purple}{\frac{19}{11}}$ respectively. If you carry the table to 265 entries, you find that $265^2 = 70225$ and $3\cdot 153^2 = 70227$ and there is your $\sqrt 3 \approx \frac{265}{153}$. You don't have to be as clever as Archimedes to think of this. No magical technique was required, no sophisticated theory was required. The calculation is tedious but straightforward and I judge that it could have been carried to 265 entries in at most a few hours, even using whatever awful technology was available at the time.

Archimedes might, of course, have used a better method. (He also produces the approximation $\frac{1351}{780}$, for which the foregoing is not obviously practical. He might have deployed the so-called “Babylonian method”. Or, having tabulated the first few approximations from the table above, he might have noticed the simple recurrence that governs them; this would not be hard for a bright high-school student. Or he might have had a some other method.) But it seems to me that since there is a straightforward and simple method by which Archimedes could have produced the required approximations, there is no puzzle to solve.

However, Rouse Ball and a number of other eminent authors disagree with me, and consider Archimedes' calculation of $\sqrt 3\approx \frac{265}{153}$ to be mysterious. (The page I linked earlier cites: T. Heath, E.T. Bell, C.B. Boyer, M. Kline, P. Beckmann, Sondheimer and Rogerson, and goes on to describe a complicated and elaborate method that Archimedes could have used.)

If it was only popular mathematics authors who thought this calculation required mysterious powers on the part of Archimedes, I would write it off as the authors overlooking the obvious, because the authors of popular books on mathematics can be quite obtuse. But Rouse Ball and Morris Kline are not dummies and I do not quite believe that they could have overlooked the method I described above, which seems not merely obvious but literally the first thing anyone would try.

My question is:

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> Is there some reason I don't appreciate that Archimedes could not have or certainly did not use the method I described? Or did all those other guys overlook the obvious?
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