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G.201 whole whole part¶

If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder is to the remainder as whole to whole.

whole_whole_part()

prove / E B / will also be to / F D / as / A B / to / C D /

assert As / A B / is to / C D /, so let the part / A E / subtracted be to the part / C F / subtracted

assert Since, as / A B / is to / C D /, so is / A E / to / C F /,

∵ magnitudes_proportional_proportionalassert Alternately also, as / A B / is to / A E /, so is / C D / to / C F /

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

assert That is, as / E B / is to / A E /, so is / F D / to / C F /, and, alternately,

∵ magnitudes_proportional_proportionalassert As / E B / is to / F D /, so is / A E / to / C F /

assert But, as / A E / is to / C F /, so by hypothesis is / A B / to / C D /

∵ ratios_ratio∴ also / E B / will be to / F D / as / A B / is to / C D /

From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional :term:`convertendo <G.200>`.

Dependency Graph¶