G.204 there magnitudes others

V.21

If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then, if ex aequali the first magnitude is greater than the third, the fourth is greater than the sixth; if equal, equal; and if less, less.

there_magnitudes_others_v_21(A, B, C, D)

given there be three magnitudes A, B, C, others D, E, F equal to them in multitude, which taken two and two are in the same ratio, let the proportion of them be perturbed, so that

prove D will also be greater than F

given there be three magnitudes A, B, C, others D, E, F equal to them in multitude, which taken two and two are in the same ratio, let the proportion of them be perturbed, so that

assert As A is to B, so is E to F,

assert as B is to C, so is D to E,

assert let A be greater than C ex aequali

assert If A = C, =

assert if less, less

∵ unequal_magnitudes_greaterassert A > C, and B is some other magnitude → therefore A has to B a greater ratio than C has to B

assert But, as A is to B, so is E to F, and, as C is to B, inversely, so is E to D

∵ magnitude_ratio_greaterassert also E has to F a greater ratio than E has to D

∵ magnitudes_ratio_greaterassert That to which the same has a greater ratio is less

assert F < D

assert D > F

assert Similarly we can prove that, if A = C, D will also = F

∴ if less, less

Dependency Graph

Required for

G.206