# Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes

### (Proposition 8 from Book 5 of Euclid's “Elements”)

For unequal magnitudes, the greater (magnitude) has a greater ratio than the lesser to the same (magnitude). And the latter (magnitude) has a greater ratio to the lesser (magnitude) than to the greater.

• Let $AB$ and $C$ be unequal magnitudes, and let $AB$ be the greater (of the two), and $D$ another random magnitude.
• I say that $AB$ has a greater ratio to $D$ than $C$ (has) to $D$, and (that) $D$ has a greater ratio to $C$ than (it has) to $AB$.

### Modern Formulation

In modern notation, this proposition reads that if $$\alpha > \beta$$ then $\frac\alpha\gamma > \frac\beta\gamma$ and $\frac\gamma\beta > \frac\gamma\alpha$

for all positive real numbers $$\alpha,\beta,\gamma$$.

### Generalized Modern Formulation

see rules of calculation with inequalities (Rules 6 and 11)

Proofs: 1

Proofs: 1 2 3 4 5
Sections: 6

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### References

#### Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

#### Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016