Relative Sizes of Components of Ratios

Theorem

As Euclid defined it:

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.

(The Elements: Book V: Proposition $14$)

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 $D$.

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 $D > B$.

$\Box$

In a similar way it can be shown that:

if $A = C$ then $B = D$

if $A < C$ then $B < D$

$\blacksquare$

Historical Note

This is Proposition 14 of Book V of Euclid's The Elements.