If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also :term:`ex aequali <G.174>` the magnitudes taken is equimultiples respectively, the one of the second and the other of the fourth.
magnitude_multiple_equimultiples()
prove/ E F / is the same multiple of B that / G H / is of D
construct Let a first magnitude A be the same multiple of a second B that a third C is of a fourth D, and let equimultiples / E F /, / G H / be taken of A, C
assert/ E F / is the same multiple of A that / G H / is of C → therefore, as many magnitudes as there are in / E F / equal to A, so many also are there in / G H / equal to C
assert Let / E F / be divided into the magnitudes / E K /, / K F /= to A, and / G H / into the magnitudes / G L /, / L H /= to C
assert Then the multitude of the magnitudes / E K /, / K F / will be equal to the multitude of the magnitudes / G L /, / L H /
assert since A is the same multiple of B that C is of D,
assert While / E K /=A, and / G L / to C,
assert/ E K / is the same multiple of B that / G L / is of D
assert The same reason
assert/ K F / is the same multiple of B that / L H / is of D
∵magnitude_multiple_fifth∴ Since, then, a first magnitude / E K / is the same multiple of a second B that a third / G L / is of a fourth D, and a fifth / K F / is also the same multiple of the second B that a sixth / L H / is of the fourth D → therefore the sum of the first and fifth, / E F /, is also the same multiple of the second B that the sum of the third and sixth, / G H /, is of the fourth D
