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G.376 commensurable magnitudes ratio¶

Commensurable magnitudes have to one another the ratio which a number has to a number.

commensurable_magnitudes_ratio(A, B)
given A, B be commensurable magnitudes
prove A has to B the ratio which a number has to a number

given A, B be commensurable magnitudes

assert A, B are commensurable, some magnitude will measure them

assert Let it measure them, and let it be C

assert as many times as C measures A, so many units let there be in D

assert as many times as C measures B, so many units let there be in E

assert then C measures A according to the units in D, while the unit also measures D according to the units in it,

assert the unit measures the number D the same number of times as the magnitude C measures A

∵ numbers_proportional_multiple_defassert as C is to A, so is the unit to D

∵ equal_magnitudes_ratioassert inversely, as A is to C, so is D to the unit

assert C measures B according to the units in E, while the unit also measures E according to the units in it,

assert the unit measures E the same number of times as C measures B

assert as C is to B, so is the unit to E

assert It was also proved that, as A is to C, so is D to the unit

∵ there_number_magnitudes_v_22∴ :term:`ex aequali <G.174>`, as A is to B, so is the number D to E

Dependency Graph¶