Those numbers are
857~=5.2,
520~=6.6,
814 ~=5.325,
540~=6.5


I think most discrepancies have to do with methods of measuring and degree of accuracy. The old books say five cells to the inch. That level of accuracy does not allow us to convert to measurements accurate to the tenth or hundredth of a millimeter. You will have some degree of sag, as well. I think the cells per square dm gives a far better picture of the density of the cells than any linear measurement. Recent studies, by the way, state that bees' size cannot be changed much via cell size, probably no more than 1%.

See: Coffey, M. F., Breen, J., Brown, M. J., & McMullan, J. B. (2010). Brood-cell size has no influence on the population dynamics of Varroa destructor mites in the native western honey bee, Apis mellifera mellifera. Apidologie, 41(5), 522-530.

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