If numbers fall between each of two numbers and an unit in continued proportion, however many numbers fall between each of them and an unit in continued proportion, so many also will fall between the numbers themselves in continued proportion.
若在两个数A、B与单位C之间分别有连比例数D、E和F、G，则A与B之间也有同样多个连比例数。
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For let the numbers D, E and F, G respectively fall between the two numbers A, B and the unit C in continued proportion; I say that, as many numbers as have fallen between each of the numbers A, B and the unit C in continued proportion, so many numbers will also fall between A, B in continued proportion. For let D by multiplying F make H, and let the numbers D, F by multiplying H make K, L respectively. Now, since, as the unit C is to the number D, so is D to E, therefore the unit C measures the number D the same number of times as D measures E. [VII. Def. 20] But the unit C measures the number D according to the units in D; therefore the number D also measures E according to the units in D; therefore D by multiplying itself has made E. Again, since, as C is to the number D, so is E to A, therefore the unit C measures the number D the same number of times as E measures A.
设D乘F得H，D、F分别乘H得K、L。由单位C与D、E、A的连比例关系，得D自乘得E，D乘E得A。
But the unit C measures the number D according to the units in D; therefore E also measures A according to the units in D; therefore D by multiplying E has made A. For the same reason also F by multiplying itself has made G, and by multiplying G has made B. And, since D by multiplying itself has made E and by multiplying F has made H, therefore, as D is to F, so is E to H. [VII. 17] For the same reason also, as D is to F, so is H to G.
同理，F自乘得G，F乘G得B。由D自乘得E且D乘F得H，根据VII.17得D比F等于E比H。
[VII. 18] Therefore also, as E is to H, so is H to G. Again, since D by multiplying the numbers E, H has made A, K respectively, therefore, as E is to H, so is A to K. [VII. 17] But, as E is to H, so is D to F; therefore also, as D is to F, so is A to K. Again, since the numbers D, F by multiplying H have made K, L respectively, therefore, as D is to F, so is K to L. [VII. 18] But, as D is to F, so is A to K; therefore also, as A is to K, so is K to L.
由D乘F得H且F自乘得G，根据VII.18得D比F等于H比G，故E比H等于H比G。由D乘E得A、D乘H得K，得E比H等于A比K，结合前式得D比F等于A比K。
Further, since F by multiplying the numbers H, G has made L, B respectively, therefore, as H is to G, so is L to B. [VII. 17] But, as H is to G, so is D to F; therefore also, as D is to F, so is L to B. But it was also proved that, as D is to F, so is A to K and K to L; therefore also, as A is to K, so is K to L and L to B. Therefore A, K, L, B are in continued proportion.
由D、F乘H得K、L，得D比F等于K比L，故A比K等于K比L。由F乘H得L、F乘G得B，得H比G等于L比B，即D比F等于L比B，结合前式得A、K、L、B成连比例。