G.198 magnitudes proportional componendo¶
magnitudes_proportional_componendo(/ E B /, / C D /, / F D /, / A B /)
given $AB$, / E B /, / C D /, / F D / be magnitudes proportional componendo, so that, as / A B / is to / E B /, so is / C D / to / F D /
prove they will also be proportional separando, that is, as / A E / is to / E B /, so is / C F / to / F D /
given $AB$, / E B /, / C D /, / F D / be magnitudes proportional componendo, so that, as / A B / is to / E B /, so is / C D / to / F D /
construct Of / A E /, / E B /, / C F /, / F D / let equimultiples / G H /, / H K /, / L M /, / M N / be taken, and of / E B /, / F D / other, chance, equimultiples, / K O /, / N P /.
∵ there_number_magnitudesassert / G H / is the same multiple of / A E / that / H K / is of / E B / → therefore / G H / is the same multiple of / A E / that / G K / is of / A B /
assert / G H / is the same multiple of / A E / that / L M / is of / C F /
assert / G K / is the same multiple of / A B / that / L M / is of / C F /
∵ there_number_magnitudesassert / L M / is the same multiple of / C F / that / M N / is of / F D / → therefore / L M / is the same multiple of / C F / that / L N / is of / C D /
assert / L M / was the same multiple of / C F / that / G K / is of / A B /
assert / G K / is the same multiple of / A B / that / L N / is of / C D /
assert / G K /, / L N / are equimultiples of / A B /, / C D /
∵ magnitude_multiple_fifthassert / H K / is the same multiple of / E B / that / M N / is of / F D /, and / K O / is also the same multiple of / E B / that / N P / is of / F D / → therefore the sum / H O / is also the same multiple of / E B / that / M P / is of / F D /
assert since, as / A B / is to / E B /, so is / C D / to / F D /, and of / A B /, / C D / equimultiples / G K /, / L N / have been taken, and of / E B /, / F D / equimultiples / H O /, / M P /,
assert if / G K / is in excess of / H O /, / L N / is also in excess of / M P /, if =, =, and if less, less
construct Let / G K / be in excess of / H O /; then, if / H K / be subtracted from each,
assert / G H / is also in excess of / K O /
assert We saw that, if / G K / was in excess of / H O /, / L N / was also in excess of / M P /
assert / L N / is also in excess of / M P /,
assert if / M N / be subtracted from each,
assert / L M / is also in excess of / N P /
assert So that, if / G H / is in excess of / K O /, / L M / is also in excess of / N P /
assert Similarly we can prove that, if / G H / = / K O /, / L M / will also = / N P /, and if less, less
assert / G H /, / L M / are equimultiples of / A E /, / C F /, while / K O /, / N P / are other, chance, equimultiples of / E B /, / F D /:
∴ as / A E / is to / E B /, so is / C F / to / F D /
Dependency Graph¶
Required for¶
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