Relative Sizes of Components of Ratios

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In the words of Euclid:

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 $\text{V}$: Proposition $14$)

That is, if $a : b = c : d$ then:

$a > c \implies b > d$
$a = c \implies b = d$
$a < c \implies b < d$


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 $D > B$.


In a similar way it can be shown that:

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


Historical Note

This proof is Proposition $14$ of Book $\text{V}$ of Euclid's The Elements.