Relative Sizes of Magnitudes on Unequal Ratios
Theorem
In the words of Euclid:
- Of magnitudes which have a ratio to the same, that which has a greater ratio is greater, and that to which the same has a greater ratio is less.
(The Elements: Book $\text{V}$: Proposition $10$)
That is:
- $a : c > b : c \implies a > b$
- $c : b > c : a \implies b < a$
Proof
Let $A$ have to $C$ a greater ratio than $B$ has to $C$.
Suppose $A = B$.
Then from Ratios of Equal Magnitudes $A : C = B : C$.
But by hypothesis $A : C > B : C$, so $A \ne B$.
Suppose $A < B$.
Then from Relative Sizes of Ratios on Unequal Magnitudes it would follow that $A : C < B : C$.
But by hypothesis $A : C > B : C$.
Therefore it must be that $A > B$.
$\Box$
Let $C$ have to $B$ a greater ratio than $C$ has to $A$.
Suppose $B = A$.
Then from Ratios of Equal Magnitudes $C : B = C : A$.
But by hypothesis $C : B > C : A$, so $B \ne A$.
Suppose $B > A$.
Then from Relative Sizes of Ratios on Unequal Magnitudes it would follow that $C : B < C : A$.
But by hypothesis $C : B > C : A$.
Therefore it must be that $B < A$.
$\blacksquare$
Historical Note
This proof is Proposition $10$ of Book $\text{V}$ of Euclid's The Elements.
It is the converse of Proposition $8$ of Book $\text{5} $: Relative Sizes of Ratios on Unequal Magnitudes.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{V}$. Propositions