I got here while trying to find a source for this myself. I didn't find the original source exactly as stated, but found what it comes from, and ordered a 1970 stat book that seems to have a section on the history of such things. It follows trivially from the approximate normalization formula given in Abdel-Aty, S. H. "Approximate Formulae for the Percentage Points and the Probability Integral of the Non-central χ 2 Distribution." Biometrika 41.3/4 (1954): 538-540.

If you want a much better approximation (for integer numbers of degrees of freedom d, down to 1, below which the chi-square distribution is not defined), try this: (d + 2*log(2) - 2/3 + 0.115668/(d + 0.1598) ) / 4^(1/d). See if you can figure out where that comes from.