If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.
若两不等量中，从大量连续减去小量，余量永不能量尽前量，则此两量不可公度。
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For, there being two unequal magnitudes AB, CD, and AB being the less, when the less is continually subtracted in turn from the greater, let that which is left over never measure the one before it; I say that the magnitudes AB, CD are incommensurable. For, if they are commensurable, some magnitude will measure them. Let a magnitude measure them, if possible, and let it be E; let AB, measuring FD, leave CF less than itself, let CF measuring BG, leave AG less than itself, and let this process be repeated continually, until there is left some magnitude which is less than E.
设两不等量AB、CD，AB较小。从大量连续减去小量，余量永不能量尽前量。
Suppose this done, and let there be left AG less than E. Then, since E measures AB, while AB measures DF, therefore E will also measure FD.
假设AB与CD可公度，则存在某量E能量尽它们。设AB量尽FD，余CF小于AB；CF量尽BG，余AG小于CF；如此重复，直至余量AG小于E。
But it measures the whole CD also; therefore it will also measure the remainder CF. But CF measures BG; therefore E also measures BG.
因E量尽AB，AB量尽DF，故E亦量尽FD。又E量尽整量CD，故E亦量尽余量CF。
But it measures the whole AB also; therefore it will also measure the remainder AG, the greater the less: which is impossible. Therefore no magnitude will measure the magnitudes AB, CD; therefore the magnitudes AB, CD are incommensurable.
因CF量尽BG，故E亦量尽BG。又E量尽AB，故E亦量尽余量AG，大量能量尽小量，不可能。故无能量尽AB、CD，即AB与CD不可公度。