, the second will be
; and in general, in the case of two incommensurable quantities, two fractions
and
can always be found, where n can be made as large as we please, one of which is less and the other greater than the true value of the ratio. For let a and b be the incommensurable quantities; then, evidently, we cannot find two multiples na, mb, such that na = mb. In this case, take any multiple of a, such as na, then this quantity must lie between some two consecutive multiples of b, such as mb, and (m + 1)b; therefore
is greater than unity, and