Book 7
36
Again, let C not measure D, and let E, the least number measured by C, D, be taken. [VII. 34]
36
Since A, B measure D, and D measures E, therefore also A, B measure E.
36
But C also measures E; therefore also A, B, C measure E.
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 E.
36
Let them measure F.
36
Since A, B, C measure F, therefore also A, B measure F; therefore the least number measured by A, B will also measure F. [VII. 35]
36
But D is the least number measured by A, B; therefore D measures F.
36
But C also measures F; therefore D, C measure F, so that the least number measured by D, C will also measure F.
36
But E is the least number measured by C, D; therefore E measures F, the greater the less: which is impossible.
36
Therefore A, B, C will not measure any number which is less than E.
36
Therefore E is the least that is measured by A, B, C. Q. E. D.
PROPOSITION 37.
37
If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number.
37
For let the number A be measured by any number B; I say that A has a part called by the same name as B.
37
For, as many times as B measures A, so many units let there be in C.
37
Since B measures A according to the units in C, and the unit D also measures the number C according to the units in it, therefore the unit D measures the number C the same number of times as B measures A.
37
Therefore, alternately, the unit D measures the number B the same number of times as C measures A; [VII. 15] therefore, whatever part the unit D is of the number B, the same part is C of A also.
37
But the unit D is a part of the number B called by the same name as it; therefore C is also a part of A called by the same name as B, so that A has a part C which is called by the same name as B. Q. E. D.
PROPOSITION 38.
38
If a number have any part whatever, it will be measured by a number called by the same name as the part.
