Book 7
4
But D is equal to each of the numbers BE, EF, FC; therefore each of the numbers BE, EF, FC is also a part of A; so that BC is parts of A.
4
Therefore etc. Q. E. D.
PROPOSITION 5.
5
If a number be a part of a number, and another be the same part of another, the sum will also be the same part of the sum that the one is of the one.
5
For let the number A be a part of BC, and another, D, the same part of another EF that A is of BC; I say that the sum of A, D is also the same part of the sum of BC, EF that A is of BC.
5
For since, whatever part A is of BC, D is also the same part of EF, therefore, as many numbers as there are in BC equal to A, so many numbers are there also in EF equal to D.
5
Let BC be divided into the numbers equal to A, namely BG, GC, and EF into the numbers equal to D, namely EH, HF; then the multitude of BG, GC will be equal to the multitude of EH, HF.
5
And, since BG is equal to A, and EH to D, therefore BG, EH are also equal to A, D.
5
For the same reason GC, HF are also equal to A, D.
5
Therefore, as many numbers as there are in BC equal to A, so many are there also in BC, EF equal to A, D.
5
Therefore, whatever multiple BC is of A, the same multiple also is the sum of BC, EF of the sum of A, D.
5
Therefore, whatever part A is of BC, the same part also is the sum of A, D of the sum of BC, EF. Q. E. D.
PROPOSITION 6.
6
If a number be parts of a number, and another be the same parts of another, the sum will also be the same parts of the sum that the one is of the one.
6
For let the number AB be parts of the number C, and another, DE, the same parts of another, F, that AB is of C; I say that the sum of AB, DE is also the same parts of the sum of C, F that AB is of C.
6
For since, whatever parts AB is of C, DE is also the same parts of F, therefore, as many parts of C as there are in AB, so many parts of F are there also in DE.
