Square Root/Examples/3/Historical Note

Historical Note on Square Root of $3$

The square root of $3$ was the second number, after the square root of $2$, to be identified as being irrational.

This was achieved by Theodorus of Cyrene.

Archimedes of Syracuse provided the approximation:

$\dfrac {1351} {780} < \sqrt 3 < \dfrac {265} {153}$

What he actually demonstrated was:

$26 - \dfrac 1 {52} < 15 \sqrt 3 < 26 - \dfrac 1 {51}$

These can be achieved by interpreting Pell's Equation, to obtain:

$1351^2 - 3 \times 780^2 = 1$

$265^2 - 3 \times 153^2 = -2$

Thus it appears that Archimedes was familiar with Pell's Equation.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 732 \, 050 \, 807 \, 568 \, 877 \, 293 \, 527 \, 446 \, 341 \, 505 \, 872 \, 366 \, 942 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 73205 \, 08075 \, 68877 \, 29352 \, 74463 \, 41505 \, 87236 \, 69428 \ldots$