Book 7
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.
39
Let D, E, F be numbers called by the same name as the parts A, B, C, and let G, the least number measured by D, E, F, be taken. [VII. 36]
39
Therefore G has parts called by the same name as D, E, F. [VII. 37]
39
But A, B, C are parts called by the same name as D, E, F; therefore G has the parts A, B, C.
39
I say next that it is also the least number that has.
39
For, if not, there will be some number less than G which will have the parts A, B, C.
39
Let it be H.
39
Since H has the parts A, B, C, therefore H will be measured by numbers called by the same name as the parts A, B, C. [VII. 38]
39
But D, E, F are numbers called by the same name as the parts A, B, C; therefore H is measured by D, E, F.
39
And it is less than G : which is impossible.
39
Therefore there will be no number less than G that will have the parts A, B, C. Q. E. D.
