Given two numbers, to investigate whether it is possible to find a third proportional to them.
给定两个数，研究是否可能找到它们的第三比例项。
给定 A、B，研究是否存在第三比例项。令 C = B·B；若 A 量 C，则得 D 使 A·D = C = B²，即 A:B = B:D；反之不可能。
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Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them. Now A, B are either prime to one another or not. And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16] Next, let A, B not be prime to one another, and let B by multiplying itself make C.
设A、B为两给定数，需研究是否可找到第三比例项。A、B要么互素，要么不互素。
Then A either measures C or does not measure it. First, let it measure it according to D; therefore A by multiplying D has made C. But, further, B has also by multiplying itself made C; therefore the product of A, D is equal to the square on B.
若互素，已证不可能找到第三比例项（IX.16）。
Therefore, as A is to B, so is B to D; [VII. 19] therefore a third proportional number D has been found to A, B. Next, let A not measure C; I say that it is impossible to find a third proportional number to A, B. For, if possible, let D, such third proportional, have been found. Therefore the product of A, D is equal to the square on B.
若不互素，令B自乘得C。若A量尽C，设量尽得D，则A乘D得C，且B自乘亦得C，故A、D之积等于B之平方，由VII.19得A:B=B:D，即找到第三比例项D。
But the square on B is C; therefore the product of A, D is equal to C. Hence A by multiplying D has made C; therefore A measures C according to D. But, by hypothesis, it also does not measure it: which is absurd.
若A量不尽C，假设存在第三比例项D，则A乘D等于B之平方即C，故A量尽C得D，与假设矛盾，故不可能。