Relative Sizes of Components of Ratios

Theorem

 * If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; if equal, equal; and if less, less.

That is, if $a : b = c : d$ then:
 * $a > c \implies b > d$
 * $a = c \implies b = d$
 * $a < c \implies b < d$

Proof
Let a first magnitude $A$ have the same ratio to a second $B$ as a third $C$ has to a fourth $D$.


 * Euclid-V-14.png

Let $A > C$.

Then from Relative Sizes of Ratios on Unequal Magnitudes $A : B > C : B$.

But $A : B = C : D$.

So from Relative Sizes of Proportional Magnitudes $C : D > C : B$.

But from Relative Sizes of Magnitudes on Unequal Ratios $D > B$.

In a similar way it can be shown that:
 * if $A = C$ then $B = D$
 * if $A < C$ then $B < D$