Book 7
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.
38
For let the number A have any part whatever, B, and let C be a number called by the same name as the part B; I say that C measures A.
38
For, since B is a part of A called by the same name as C, and the unit D is also a part of C called by the same name as it, therefore, whatever part the unit D is of the number C, the same part is B of A also; therefore the unit D measures the number C the same number of times that B measures A.
38
Therefore, alternately, the unit D measures the number B the same number of times that C measures A. [VII. 15]
38
Therefore C measures A. Q. E. D.
PROPOSITION 39.
39
To find the number which is the least that will have given parts.
39
Let A, B, C be the given parts; thus it is required to find the number which is the least that will have the parts A, B, C.
