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 $B$.

fig07e

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