In Part II we considered the problem of measuring the length of line segments—or, to be more precise, we considered the problem of comparing two line segments AB, CD by looking for a common measure. We discovered that a given pair of line segments AB, CD may have no common measure at all. However in this case, though our procedure for finding common measures could not possibly produce a genuine common measure as it was designed to do, it could nevertheless be interpreted to obtain a value for, or a sequence of better and better approximations to, the irrational ratio AB/CD (see Chapter II.13).
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