I have been working my way through Leslie Stroebel's "View Camera Technique", 7th Edition, and have now come across some calculations for which I couldn't follow his train of thought, and cannot mathematically reproduce what he has written. The section in question is the Intermediate f-number calculations. He writes (p.71):

"... For 1/2 stops the factor is the square root of 1.5 (1.22). For 1/3 stops, the factor is the square root of 1.33 (1.15), and for 2/3 stops the factor is the square root of 1.67 (1.29). Multiplying f/8, for example, by these factors to determine f-numbers that represent stopping down by 1/3, 1/2, 2/3, and one stop, the f-numbers are f/9.2, f/9.76, f/10.32, and f/11. The factor for 1/10 stop is the square root of 1.1 (1.049)."

I take it that most know that to get one full stop, one multiplies by the sqrt(2) = 1.41, e.g. f/5.6 * 1.41 = f/8. The approximate nature of the old sequence 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64... makes it hard to verify, but it does seem to work out if you go high enough. Each number is a little off, but the sequence works out.

I would think that the factors that he is talking about should be usable to go from any steps to any subsequent step, and thus can also be used repeatedly to go through all steps. The problem is that these numbers do not work like that, except the first one, for obvious reasons. Using full accuracy in my calculator, 1.22*1.22=1.5, not 1.41 (duh), 1.15*1.15*1.15=1.5396, not 1.41. I don't know why he calculates 2/3 stop directly (and wrongly), rather than just multiplying the factor for 1/3 stops by itself, but even so, 1.15*1.15 (with full accuracy) is 1.333333 (of course), not 1.29. Multiplying his calculated value for 1/10th stop by itself 10 times gives 1.61, not 1.41.

I believe that what he should do is to take the nth root of sqrt(2) to get the nth fraction of a stop. For example, to get half stops, he should take the root of sqrt(2) = 1.189. Multiplying this number by itself would give sqrt(2), i.e. 2 half stops equals 1 stop. 1/3 stops would then be the 3rd root of sqrt(2) = 1.122, 1/10th stops would be the 10th root of sqrt(2) = 1.0352, and so on. I won't calculate 2/3 stops His sequence from f/8 to f/11 should be: f/8, f/8.97, f/9.51, f/10.07, f/11. Note that f/11 should actually be f/11.3, if one takes f/8 to be exact, part of the problem with the rounding in the sequence, and verification.

Has someone else come across this, or can someone explain to me where I went wrong?