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1 Here was another diagram （one would expect the sentence to run ‘Let two lines representing . . . have been similarly divided’）. Two segments, A and B, of one line represented two persons, two segments, C and D, of another their shares. It is shown that, if A:B::C:D, then A+C:B+D::A:B, i.e., if the shares are proportioned to the persons, their relative condition after receiving them will be the same as it was before.