| To find the least number which three given numbers measure. | ||
| Let A, B, and C be the three given numbers.
It is required to find the least number which they measure. |
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| Take D the least number measured by the two numbers A and B. | VII.34 | |
| Then C either measures, or does not measure, D.
First, let it measure it. But A and B also measure D, therefore A, B, and C measure D. I say next that it is also the least that they measure. |
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| If not, A, B, and C measure some number E less than D. | ||
| Since A, B, and C measure E, therefore A and B measure E. Therefore the least number measured by A and B also measures E. | VII.35 | |
| But D is the least number measured by A and B, therefore D measures E, the greater the less, which is impossible.
Therefore A, B, and C do not measure any number less than D. Therefore D is the least that A, B, and C measure. Next, let C not measure D. | ||
| Take E, the least number measured by C and D. | VII.34 | |
| Since A and B measure D, and D measures E, therefore A and B also measure E. But C also measures E, therefore A, B, and C also measure E.
I say next that it is also the least that they measure. If not, A, B, and C measure some number F less than E. |
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| Since A, B, and C measure F, therefore A and B measure F. Therefore the least number measured by A and B also measures F. But D is the least number measured by A and B, therefore D measures F. But C also measures F, therefore D and C measure F, so that the least number measured by D and C also measures F. | VII.35 | |
| But E is the least number measured by C and D, therefore E measures F, the greater the less, which is impossible.
Therefore A, B, and C do not measure any number which is less than E. Therefore E is the least that is measured by A, B, and C. | ||
| Q.E.D. | ||
Book VII Introduction - Proposition VII.35 - Proposition VII.37.