If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.
Let a first magnitude, / A B /, be the same multiple of a second, C, that a third, / D E /, is of a fourth, F, and let a fifth, / B G /, also be the same multiple of the second, C, that a sixth, / E H /, is of the fourth F;
I say that the sum of the first and fifth, / A G /, will be the same multiple of the second, C, that the sum of the third and sixth, / D H /, is of the fourth, F.
For, since / A B / is the same multiple of C that / D E / is of F,
therefore, as many magnitudes as there are in / A B / equal to C, so many also are there in / D E / equal to F.
For the same reason also,
as many as there are in / B G / equal to C, so many are there also in / E H / equal to F;
therefore, as many as there are in the whole / A G / equal to C, so many also are there in the whole / D H / equal to F.
Therefore, whatever multiple / A G / is of C, that multiple also is / D H / of F.
Therefore the sum of the first and fifth, / A G /, is the same multiple of the second, C, that the sum of the third and sixth, / D H /, is of the fourth, F.
