Gene asks:

<< If x is irrational, and I wish to best approximate the ratio x:1 by integers

a:b, then I would use continued fractions. Suppose x and y are irrational,

and I wish to best approximate the ratio x:y:1 by integers a:b:c. Is it known how to do this?

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Depends what you mean by 'best'. For two irrationals, you might want to look into cubic continued fractions. Here are two papers on bifurcating continued fractions that give good approximations; the determinant of three consecutive approximations is 1 (whereas normal CF's give +/-1), so in that sense they're optimal. (arxiv.org/pdf/math.GM/0002227 and /0008060) -- Mike Stay staym@clear.net.nz -- Mike Stay staym@clear.net.nz