Given three commensurable magnitudes, to find their greatest common measure.
给定三个可公度的量，求它们的最大公度。
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Let A, B, C be the three given commensurable magnitudes; thus it is required to find the greatest common measure of A, B, C. Let the greatest common measure of the two magnitudes A, B be taken, and let it be D; [X. 3] then D either measures C, or does not measure it. First, let it measure it. Since then D measures C, while it also measures A, B, therefore D is a common measure of A, B, C. And it is manifest that it is also the greatest; for a greater magnitude than the magnitude D does not measure A, B. Next, let D not measure C.
取A、B的最大公度D。若D量尽C，则D即为A、B、C的最大公度。
I say first that C, D are commensurable. For, since A, B, C are commensurable, some magnitude will measure them, and this will of course measure A, B also; so that it will also measure the greatest common measure of A, B, namely D. [X. 3, Por.] But it also measures C; so that the said magnitude will measure C, D; therefore C, D are commensurable. Now let their greatest common measure be taken, and let it be E. [X. 3] Since then E measures D, while D measures A, B, therefore E will also measure A, B.
若D不量尽C，则C与D可公度，因为存在某量同时量尽A、B、C，从而也量尽D和C。
But it measures C also; therefore E measures A, B, C; therefore E is a common measure of A, B, C. I say next that it is also the greatest. For, if possible, let there be some magnitude F greater than E, and let it measure A, B, C. Now, since F measures A, B, C, it will also measure A, B, and will measure the greatest common measure of A, B. [X. 3, Por.] But the greatest common measure of A, B is D; therefore F measures D. But it measures C also; therefore F measures C, D; therefore F will also measure the greatest common measure of C, D.
取C与D的最大公度E。因E量尽D，而D量尽A、B，故E量尽A、B；又E量尽C，故E是A、B、C的公度。
[X. 3, Por.] But that is E; therefore F will measure E, the greater the less: which is impossible. Therefore no magnitude greater than the magnitude E will measure A, B, C; therefore E is the greatest common measure of A, B, C if D do not measure C, and, if it measure it, D is itself the greatest common measure. Therefore the greatest common measure of the three given commensurable magnitudes has been found. PORISM. From this it is manifest that, if a magnitude measure three magnitudes, it will also measure their greatest common measure.
若存在大于E的量F量尽A、B、C，则F量尽D和C，从而量尽E，矛盾。故E是最大公度。