# Proposition: 5.16: Proportional Magnitudes are Proportional Alternately

### (Proposition 16 from Book 5 of Euclid's “Elements”)

If four magnitudes are proportional then they will also be proportional alternately.

• Let $A$, $B$, $C$ and $D$ be four proportional magnitudes, (such that) as $A$ (is) to $B$, so $C$ (is) to $D$.
• I say that they will also be [proportional] alternately, (so that) as $A$ (is) to $C$, so $B$ (is) to $D$.

### Modern Formulation

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

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

### Generalized Modern Formulation

This is a special case of the fact that the multiplication of real numbers is cancellative. Since for $\frac\beta\gamma\neq 0$:

$$\frac\alpha\beta=\frac\gamma\delta\Leftrightarrow \frac \beta\gamma \frac\alpha\beta=\frac \beta\gamma \frac\gamma\delta.$$

Proofs: 1

Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Sections: 27

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

Github:

non-Github:
@Fitzpatrick