Recited from memory by: unstattedCommoner
The author writes:
The approximation sqrt(2) ~= 665857/470832 was obtained by carrying out three iterations of the recurrence relation
a(n+1) = a(n)2 + 2b(n)2 b(n+1) = 2a(n)b(n)
starting with a(0) = 3 and b(0) = 2. x(n) = a(n)/b(n) is then the approximation to sqrt(2).
Convergence of this method is extremely rapid:
x(n) - sqrt(2) is roughly proportional to ((x(0)-sqrt(2))/(2 sqrt(2)))(2n).