# Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios

### Euclid's Formulation

For (magnitudes) having a ratio to the same (magnitude), that > For (magnitudes) having a ratio to the same (magnitude), that (magnitude which) has the greater ratio is (the) greater. And that (magnitude) to which the latter (magnitude) has a greater ratio is (the) lesser. * For let $A$ have a greater ratio to $C$ than $B$ (has) to $C$. * I say that $A$ is greater than $B$.

### Modern Formulation

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

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

### Generalized Modern Formulation

see rules of calculation with inequalities (Rules 6)

Proofs: 1

Proofs: 1 2 3
Sections: 4

Thank you to the contributors under CC BY-SA 4.0!

Github:

non-Github:
@Fitzpatrick

### 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