Given three numbers not prime to one another, to find their greatest common measure.
给定三个互不互质的数，求它们的最大公度数。
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Let A, B, C be the three given numbers not prime to one another; thus it is required to find the greatest common measure of A, B, C. For let the greatest common measure, D, of the two numbers A, B be taken; [VII. 2] then D either measures, or does not measure, C. First, let it measure it. But it measures A, B also; therefore D measures A, B, C; therefore D is a common measure of A, B, C. I say that it is also the greatest. For, if D is not the greatest common measure of A, B, C, some number which is greater than D will measure the numbers A, B, C.
设A、B、C为三个互不互质的数，先取A、B的最大公度数D。
Let such a number measure them, and let it be E. Since then E measures A, B, C, it will also measure A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.] But the greatest common measure of A, B is D; therefore E measures D, the greater the less: which is impossible. Therefore no number which is greater than D will measure the numbers A, B, C; therefore D is the greatest common measure of A, B, C. Next, let D not measure C; I say first that C, D are not prime to one another. For, since A, B, C are not prime to one another, some number will measure them.
若D量尽C，则D是A、B、C的公度数。假设存在大于D的数E能量尽A、B、C，则E能量尽A、B，从而能量尽D，矛盾。故D为最大。
Now that which measures A, B, C will also measure A, B, and will measure D, the greatest common measure of A, B. [VII. 2, Por.] But it measures C also; therefore some number will measure the numbers D, C; therefore D, C are not prime to one another. Let then their greatest common measure E be taken. [VII. 2] Then, since E measures D, and D measures A, B, therefore E also measures A, B. 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.
若D不量尽C，则D与C不互质。取D、C的最大公度数E。
For, if E is not the greatest common measure of A, B, C, some number which is greater than E will measure the numbers A, B, C. Let such a number measure them, and let it be F. Now, since F measures A, B, C, it also measures A, B; therefore it will also measure the greatest common measure of A, B. [VII. 2, Por.] But the greatest common measure of A, B is D; therefore F measures D. And it measures C also; therefore F measures D, C; therefore it will also measure the greatest common measure of D, C. [VII. 2, Por.] But the greatest common measure of D, C is E; therefore F measures E, the greater the less: which is impossible.
由于E量尽D，D量尽A、B，故E量尽A、B；又E量尽C，故E是A、B、C的公度数。假设存在大于E的数F能量尽A、B、C，则F量尽A、B，从而量尽D；又F量尽C，故F量尽D、C，从而量尽E，矛盾。故E为最大。