# Proposition: 5.07: Ratios of Equal Magnitudes

### Euclid's Formulation

Equal (magnitudes) have the same ratio to the same (magnitude), and the latter (magnitude has the same ratio) to the equal (magnitudes).

• Let $A$ and $B$ be equal magnitudes, and $C$ some other random magnitude.
• I say that $A$ and $B$ each have the same ratio to $C$, and (that) $C$ (has the same ratio) to each of $A$ and $B$.

### Modern Formulation

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

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

### Generalized Formulation

The above proposition is even true for all real numbers with $$\alpha\neq 0, \beta\neq 0, \gamma\neq 0$$. Algebraically, it follows from the existence and uniqueness of inverse real numbers with respect to multiplication.

Proofs: 1 Corollaries: 1

Proofs: 1 2 3 4 5 6 7 8 9 10
Sections: 11

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