If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will be prime to the remaining number.
若三个数成连比例，且是与它们同比的数中最小的，则其中任意两个数的和与剩下的一个数互素。
三连比例 A、B、C 是同比中最小；取最小数 DE、EF（在同一直线上，DF = DE + EF）：DE 自乘 = A，DE·EF = B，EF 自乘 = C。DE、EF 互素 ⇒ 任意两和与第三互素。
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Let A, B, C, three numbers in continued proportion, be the least of those which have the same ratio with them; I say that any two of the numbers A, B, C whatever added together are prime to the remaining number, namely A, B to C; B, C to A; and further A, C to B. For let two numbers DE, EF, the least of those which have the same ratio with A, B, C, be taken. [VIII. 2] It is then manifest that DE by multiplying itself has made A, and by multiplying EF has made B, and, further, EF by multiplying itself has made C. [VIII. 2] Now, since DE, EF are least, they are prime to one another.
取与A、B、C同比的最小数DE、EF，则DE自乘得A，DE乘EF得B，EF自乘得C。
[VII. 22] But, if two numbers be prime to one another, their sum is also prime to each; [VII. 28] therefore DF is also prime to each of the numbers DE, EF. But further DE is also prime to EF; therefore DF, DE are prime to EF. But, if two numbers be prime to any number, their product is also prime to the other; [VII. 24] so that the product of FD, DE is prime to EF; hence the product of FD, DE is also prime to the square on EF. [VII. 25] But the product of FD, DE is the square on DE together with the product of DE, EF; [II. 3] therefore the square on DE together with the product of DE, EF is prime to the square on EF.
由于DE、EF互素，其和DF与DE、EF均互素，故DF与DE均与EF互素，从而FD与DE的积与EF互素，进而与EF的平方互素。
And the square on DE is A, the product of DE, EF is B, and the square on EF is C; therefore A, B added together are prime to C. Similarly we can prove that B, C added together are prime to A. I say next that A, C added together are also prime to B. For, since DF is prime to each of the numbers DE, EF, the square on DF is also prime to the product of DE, EF.
FD与DE的积等于DE的平方加DE乘EF，即A加B，故A加B与C互素。同理可证B加C与A互素。
[VII. 24, 25] But the squares on DE, EF together with twice the product of DE, EF are equal to the square on DF; [II. 4] therefore the squares on DE, EF together with twice the product of DE, EF are prime to the product of DE, EF. Separando, the squares on DE, EF together with once the product of DE, EF are prime to the product of DE, EF. Therefore, separando again, the squares on DE, EF are prime to the product of DE, EF. And the square on DE is A, the product of DE, EF is B, and the square on EF is C.
因DF与DE、EF均互素，DF的平方与DE乘EF互素。而DF的平方等于DE、EF的平方和加二倍DE乘EF，故DE、EF的平方和加二倍DE乘EF与DE乘EF互素，分离得DE、EF的平方和与DE乘EF互素，即A加C与B互素。