Given three numbers, to investigate when it is possible to find a fourth proportional to them.
给定三个数，研究何时可以找到它们的第四比例数。
给定 A、B、C，研究何时存在第四比例项 D；分四种情况讨论首末是否互质、是否成连比例，引入辅助 E。
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Let A, B, C be the given three numbers, and let it be required to investigate when it is possible to find a fourth proportional to them. Now either they are not in continued proportion, and the extremes of them are prime to one another; or they are in continued proportion, and the extremes of them are not prime to one another; or they are not in continued proportion, nor are the extremes of them prime to one another; or they are in continued proportion, and the extremes of them are prime to one another. If then A, B, C are in continued proportion, and the extremes of them A, C are prime to one another, it has been proved that it is impossible to find a fourth proportional number to them. [IX. 17] <*>Next, let A, B, C not be in continued proportion, the extremes being again prime to one another; I say that in this case also it is impossible to find a fourth proportional to them. For, if possible, let D have been found, so that, as A is to B, so is C to D, and let it be contrived that, as B is to C, so is D to E. Now, since, as A is to B, so is C to D, and, as B is to C, so is D to E, therefore, ex aequali, as A is to C, so is C to E.
若三数成连比例且两端互素，则据IX.17，不可能找到第四比例数。
[VII. 14] But A, C are prime, primes are also least, [VII. 21] and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. [VII. 20] Therefore A measures C as antecedent antecedent. But it also measures itself; therefore A measures A, C which are prime to one another: which is impossible. Therefore it is not possible to find a fourth proportional to A, B, C. <*> Next, let A, B, C be again in continued proportion, but let A, C not be prime to one another. I say that it is possible to find a fourth proportional to them.
若三数不成连比例但两端互素，假设存在第四比例数D，则通过构造E并利用比例性质推出A度量互素的A、C，矛盾。
For let B by multiplying C make D; therefore A either measures D or does not measure it. First, let it measure it according to E; therefore A by multiplying E has made D. But, further, B has also by multiplying C made D; therefore the product of A, E is equal to the product of B, C; therefore, proportionally, as A is to B, so is C to E; [VII. 19] therefore E has been found a fourth proportional to A, B, C. Next, let A not measure D; I say that it is impossible to find a fourth proportional number to A, B, C. For, if possible, let E have been found; therefore the product of A, E is equal to the product of B, C. [VII. 19] But the product of B, C is D; therefore the product of A, E is also equal to D.
若三数成连比例且两端不互素，令B乘C得D。若A度量D，则存在E使A乘E等于D，从而A:B=C:E，E即为所求。
Therefore A by multiplying E has made D; therefore A measures D according to E, so that A measures D. But it also does not measure it: which is absurd. Therefore it is not possible to find a fourth proportional number to A, B, C when A does not measure D. Next, let A, B, C not be in continued proportion, nor the extremes prime to one another. And let B by multiplying C make D.
若A不度量D，假设存在第四比例数E，则A乘E等于B乘C即D，故A度量D，与假设矛盾，故不可能。