There's actually a really sneaky moment in here which is not totally revealed. Feynman says that he first works out 12, then 12.002, then the man with the abacu

There's actually a really sneaky moment in here which is not totally revealed. Feynman says that he first works out 12, then 12.002, then the man with the abacus says "12.01!". He asks for more digits. What is sneaky here is that 12.002 is a very rough approximation that Feynman is making before doing a long division. He has the formula of 12 * (1 + 1.03/1728)^(1/3) ~= 12 * (1 + (1/3) * 1.03/1728) But he simply hasn't had the time to do the long division yet and has just approximated 1.03 / 1728 by 1/2000. At this point he has the formula, and quite possibly even knows that the error goes like - 12 x^2 / 9 which would be something like one part in a million, but he wants to do his long division to get the extra decimals to show off. There's actually a slightly more slick way to get to this where you start from 12 and compute: (2/3) * guess + (1/3) * number / guess^2. The 1/3 and 2/3 are chosen to minimize the error, and you wouldn't know this if you've never worked it out. (I only worked it out because I would occasionally be stuck without a calculator on exam problems.) It works really well on these sorts of problems; you compute 1729.03 / 144 and aside from the leading 12 you get: 0.0071527777... You divide this fraction by 3 to get: 0.002384259259... which is precisely as far as Feynman got, but the reasoning is much quicker. If he'd been even faster with this he might have been able to apply guess = 12.002 to get another couple of decimals in the same time.