| To find the greatest common measure of three given commensurable magnitudes. | ||
| Let A, B, and C be the three given commensurable magnitudes.
It is required to find the greatest common measure of A, B, and C. | ||
| Take the greatest common measure D of the two magnitudes A and B. | X.3 | |
| Either D measures C, or it does not measure it. | ||
| First, let it measure it.
Since then D measures C, while it also measures A and B, therefore D is a common measure of A, B, and C. And it is manifest that it is also the greatest, for a greater magnitude than the magnitude D does not measure A and B. |
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| Next, let D not measure C.
I say first that C and D are commensurable. | ||
| Since A, B, and C are commensurable, some magnitude measures them, and this of course measures A and B also, so that it also measures the greatest common measure of A and B, namely D. | X.3.Cor. | |
| But it also measures C, so that the said magnitude measures C and D, therefore C and D are commensurable. | ||
| Now take their greatest common measure E. | X.3 | |
| Since E measures D, while D measures A and B, therefore E also measures A and B. But it measures C also, therefore E measures A, B, and C. Therefore E is a common measure of A, B, and 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, and C. | ||
| Now, since F measures A, B, and C, it also measures A and B, and therefore measures the greatest common measure of A and B. | X.3,Cor. | |
| But the greatest common measure of A and B is D, therefore F measures D.
But it measures C also, therefore F measures C and D. Therefore F also measures the greatest common measure of C and D. But that is E, therefore F measures E, the greater the less, which is impossible. |
X.3,Cor. | |
| Therefore no magnitude greater than the magnitude E measures A, B, and C. Therefore E is the greatest common measure of A, B, and C if D does not measure C, but if it measures it, then D is itself the greatest common measure.
Therefore the greatest common measure of the three given commensurable magnitudes has been found. | ||
Corollary.From this it is manifest that, if a magnitude measures three magnitudes, then it also measures their greatest common measure. The greatest common measure can be found similarly for more magnitudes, and the corollary extended. | ||
| Q.E.D. | ||
