G.207 magnitude ratio fifth

V.24

If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.

magnitude_ratio_fifth(/ A B /, C, / D E /, F)

given a first magnitude / A B / have to a second C the same ratio as a third / D E / has to a fourth F

prove the first and fifth added together, / A G /, will have to the second C the same ratio as the third and sixth, / D H /, has to the fourth F

given a first magnitude / A B / have to a second C the same ratio as a third / D E / has to a fourth F

assert let also a fifth / B G / have to the second C the same ratio as a sixth / E H / has to the fourth F

assert Since, as / B G / is to C, so is / E H / to F,

assert Inversely, as C is to / B G /, so is F to / E H /

assert Since, then, as / A B / is to C, so is / D E / to F, and, as C is to / B G /, so is F to / E H /,

∵ there_number_magnitudes_v_22assert :term:`ex aequali <G.174>`, as / A B / is to / B G /, so is / D E / to / E H /

∵ magnitudes_proportional_separandoassert since the magnitudes are proportional :term:`separando <G.197>`, they will also be proportional :term:`componendo <G.196>`

assert as / A G / is to / B G /, so is / D H / to / E H /

assert Also, as / B G / is to C, so is / E H / to F

∵ there_number_magnitudes_v_22∴ ex aequali, as / A G / is to C, so is / D H / to F

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