# Proposition: 5.13: Relative Sizes of Proportional Magnitudes

### (Proposition 13 from Book 5 of Euclid's “Elements”)

If a first (magnitude) has the same ratio to a second that a third (has) to a fourth, and the third (magnitude) has a greater ratio to the fourth than a fifth (has) to a sixth, then the first (magnitude) will also have a greater ratio to the second than the fifth (has) to the sixth. * For let a first (magnitude) $A$ have the same ratio to a second $B$ that a third $C$ (has) to a fourth $D$, and let the third (magnitude) $C$ have a greater ratio to the fourth $D$ than a fifth $E$ (has) to a sixth $F$. * I say that the first (magnitude) $A$ will also have a greater ratio to the second $B$ than the fifth $E$ (has) to the sixth $F$.

### Modern Formulation

In modern notation, this proposition reads that if $\frac\alpha\beta=\frac\gamma\delta\text{ and }\frac\gamma\delta > \frac\epsilon\zeta$ then $\frac\alpha\beta > \frac\epsilon\zeta$

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

### Generalized Modern Formulation

see rules of calculation with inequalities (Rule 11)

Proofs: 1

Proofs: 1 2 3
Sections: 4

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