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