Book 7
34
But F, E are least, and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less; [VII. 20] therefore E measures G.
34
And, since A by multiplying E, G has made C, D, therefore, as E is to G, so is C to D. [VII. 17]
34
But E measures G; therefore C also measures D, the greater the less: which is impossible.
34
Therefore A, B will not measure any number which is less than C.
34
Therefore C is the least that is measured by A, B. Q. E. D.
PROPOSITION 35.
35
If two numbers measure any number, the least number measured by them will also measure the same.
35
For let the two numbers A, B measure any number CD, and let E be the least that they measure; I say that E also measures CD.
35
For, if E does not measure CD, let E, measuring DF, leave CF less than itself.
35
Now, since A, B measure E, and E measures DF, therefore A, B will also measure DF.
35
But they also measure the whole CD; therefore they will also measure the remainder CF which is less than E: which is impossible.
35
Therefore E cannot fail to measure CD; therefore it measures it. Q. E. D.
PROPOSITION 36.
36
Given three numbers, to find the least number which they measure.
36
Let A, B, C be the three given numbers; thus it is required to find the least number which they measure.
36
Let D, the least number measured by the two numbers A, B, be taken. [VII. 34]
36
Then C either measures, or does not measure, D.
36
First, let it measure it.
36
But A, B also measure D; therefore A, B, C measure D.
36
I say next that it is also the least that they measure.
36
For, if not, A, B, C will measure some number which is less than D.
36
Let them measure E.
36
Since A, B, C measure E, therefore also A, B measure E.
36
Therefore the least number measured by A, B will also measure E. [VII. 35]
36
But D is the least number measured by A, B; therefore D will measure E, the greater the less: which is impossible.
36
Therefore A, B, C will not measure any number which is less than D; therefore D is the least that A, B, C measure.
