If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.
there_number_magnitudes(/ A B /, / C D /, E, F)
given any number of magnitudes whatever / A B /, / C D / be equimultiples of any magnitudes E, F equal in multitude
prove whatever multiple / A B / is of E, that multiple will / A B /, / C D / also be of E, F
given any number of magnitudes whatever / A B /, / C D / be equimultiples of any magnitudes E, F equal in multitude
assert I say that, whatever multiple / A B / is of E, that multiple will / A B /, / C D / also be of E, F
assert/ A B / is the same multiple of E that / C D / is of F, as many magnitudes as there are in / A B / equal to E, so many also are there in / C D / equal to F
assert Let / A B / be divided into the magnitudes / A G /, / G B /= to E, and / C D / into / C H /, / H D /= to F
assert Then the multitude of the magnitudes / A G /, / G B / will be equal to the multitude of the magnitudes / C H /, / H D /
assert/ A G /=E, and / C H / to F → therefore / A G /=E, and / A G /, / C H / to E, F
assert The same reason
assert/ G B /=E, and / G B /, / H D / to E, F
assert as many magnitudes as there are in / A B / equal to E, so many also are there in / A B /, / C D / equal to E, F
∴ whatever multiple / A B / is of E, that multiple will / A B /, / C D / also be of E, F
