A Disquisition on the Square Root of Three - Continued Fractions

A "standard" sequence of continued fractions for approximating $\sqrt{3}$ follows.

$$1+\frac{2}{2}=2=2.000000$$

$$1+\frac{2}{2+\frac{2}{2}}=\frac{5}{3}\approx 1.666667$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}=\frac{7}{4}=1.750000$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}=\frac{19}{11}\approx 1.727273$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}=\frac{26}{15}\approx 1.733333$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}=\frac{71}{41}\approx 1.731707$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}=\frac{97}{56}\approx 1.732143$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}=\frac{265}{153}\approx 1.732026$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}=\frac{362}{209}\approx 1.732057$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}}=\frac{989}{571}\approx 1.732049$$

$$1+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2+\frac{2}{2}}}}}}}}}}}=\frac{1351}{780}\approx 1.732051$$

These approximations give the same fractions as the Greek ladder table. Thus, it seems fair to declare that the continued fraction method of estimating $\sqrt 3$ ties the Greek ladder method.