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Proposition 36

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.
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. |
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. |
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.

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This proposition is used in the proof of proposition VII.39.

Book VII Introduction - Proposition VII.35 - Proposition VII.37.

© 1996
D.E.Joyce
Clark University