But it is also less: which is impossible. Therefore, as the square on BD is to the square on FH, so is not the circle ABCD to any area less than the circle EFGH. Similarly we can prove that neither is the circle EFGH to any area less than the circle ABCD as the square on FH is to the square on BD. I say next that neither is the circle ABCD to any area greater than the circle EFGH as the square on BD is to the square on FH. For, if possible, let it be in that ratio to a greater area S. Therefore, inversely, as the square on FH is to the square on DB, so is the area S to the circle ABCD. But, as the area S is to the circle ABCD, so is the circle EFGH to some area less than the circle ABCD; therefore also, as the square on FH is to the square on BD, so is the circle EFGH to some area less than the circle ABCD: [V. 11] which was proved impossible. Therefore, as the square on BD is to the square on FH, so is not the circle ABCD to any area greater than the circle EFGH. And it was proved that neither is it in that ratio to any area less than the circle EFGH; therefore, as the square on BD is to the square on FH, so is the circle ABCD to the circle EFGH. Therefore etc. Q. E. D.
But it is also less: which is impossible. Therefore, as the square on BD is to the square on FH, so is not the circle ABCD to any area less than the circle EFGH. Similarly we can prove that neither is the circle EFGH to any area less than the circle ABCD as the square on FH is to the square on BD. I say next that neither is the circle ABCD to any area greater than the circle EFGH as the square on BD is to the square on FH. For, if possible, let it be in that ratio to a greater area S. Therefore, inversely, as the square on FH is to the square on DB, so is the area S to the circle ABCD. But, as the area S is to the circle ABCD, so is the circle EFGH to some area less than the circle ABCD; therefore also, as the square on FH is to the square on BD, so is the circle EFGH to some area less than the circle ABCD: [V. 11] which was proved impossible. Therefore, as the square on BD is to the square on FH, so is not the circle ABCD to any area greater than the circle EFGH. And it was proved that neither is it in that ratio to any area less than the circle EFGH; therefore, as the square on BD is to the square on FH, so is the circle ABCD to the circle EFGH. Therefore etc. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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