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 is the same multiple of the second that the sum of the third and sixth is of the fourth.
magnitude_multiple_fifth(/ A B /, C, / D E /, F)
given a first magnitude, / A B /, be the same multiple of a second, C, that a third, / D E /, is of a fourth, F, let a fifth, / B G /, also be the same multiple of the second, C, that a sixth, / E H /, is of the fourth F
prove 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
given a first magnitude, / A B /, be the same multiple of a second, C, that a third, / D E /, is of a fourth, F, let a fifth, / B G /, also be the same multiple of the second, C, that a sixth, / E H /, is of the fourth F
assert/ 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
assert The same reason also,
assert As many as there are in / B G / equal to C, so many are there also in / E H / equal to F
assert as many as there are in / A G / equal to C, so many also are there in / D H / equal to F
assert whatever multiple / A G / is of C, that multiple also is / D H / of F
∴ 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
