For since, whatever parts AB is of C, the same parts also is DE of F,
therefore, as many parts of C as there are in AB, so many parts also of F are there in DE.
Let AB be divided into the parts of C, namely AG, GB, and DE into the parts of F, namely DH, HE;
thus the multitude of AG, GB will be equal to the multitude of DH, HE.
Now since, whatever part AG is of C, the same part also is DH of F,
alternately also, whatever part or parts AG is of DH, the same part or the same parts is C of F also. [VII. 9]
For the same reason also, whatever part or parts GB is of HE, the same part or the same parts is C of F also;
so that,
in addition, whatever parts or part AB is of DE, the same parts also, or the same part, is C of F. [VII. 5, 6]
Q. E. D.
Complete text
Book I