Given two commensurable magnitudes, to find their greatest common measure.
给定两个可公度的量，求它们的最大公度。
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Let the two given commensurable magnitudes be AB, CD of which AB is the less; thus it is required to find the greatest common measure of AB, CD. Now the magnitude AB either measures CD or it does not. If then it measures it—and it measures itself also—AB is a common measure of AB, CD. And it is manifest that it is also the greatest; for a greater magnitude than the magnitude AB will not measure AB. Next, let AB not measure CD. Then, if the less be continually subtracted in turn from the greater, that which is left over will sometime measure the one before it, because AB, CD are not incommensurable; [cf. X. 2] let AB, measuring ED, leave EC less than itself, let EC, measuring FB, leave AF less than itself, and let AF measure CE.
设两可公度量AB和CD，AB较小。若AB量尽CD，则AB是公度，且因大于AB的量不能量尽AB，故AB为最大公度。
Since, then, AF measures CE, while CE measures FB, therefore AF will also measure FB. But it measures itself also; therefore AF will also measure the whole AB. But AB measures DE; therefore AF will also measure ED. But it measures CE also; therefore it also measures the whole CD. Therefore AF is a common measure of AB, CD. I say next that it is also the greatest.
若AB不量尽CD，则从较大者中连续减去较小者，余量最终能量尽前一个量。设AB量尽ED余EC小于AB，EC量尽FB余AF小于EC，且AF量尽CE。
For, if not, there will be some magnitude greater than AF which will measure AB, CD. Let it be G. Since then G measures AB, while AB measures ED, therefore G will also measure ED. But it measures the whole CD also; therefore G will also measure the remainder CE. But CE measures FB; therefore G will also measure FB. But it measures the whole AB also, and it will therefore measure the remainder AF, the greater the less: which is impossible.
因AF量尽CE，CE量尽FB，故AF量尽FB；又AF量尽自身，故AF量尽AB。AB量尽DE，故AF量尽ED；又AF量尽CE，故AF量尽CD。因此AF是AB和CD的公度。
Therefore no magnitude greater than AF will measure AB, CD; therefore AF is the greatest common measure of AB, CD. Therefore the greatest common measure of the two given commensurable magnitudes AB, CD has been found. Q. E. D. PORISM.
若存在大于AF的量G量尽AB和CD，则G量尽ED和CD，故G量尽CE；又CE量尽FB，故G量尽FB；又G量尽AB，故G量尽余量AF，大数量尽小量，不可能。故AF是最大公度。