# Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated

### (Proposition 17 from Book 5 of Euclid's “Elements”)

If composed magnitudes are proportional then they will also be proportional (when) separated. * Let $AB$, $BE$, $CD$, and $DF$ be composed magnitudes (which are) proportional, (so that) as $AB$ (is) to $BE$, so $CD$ (is) to $DF$. * I say that they will also be proportional (when) separated, (so that) as $AE$ (is) to $EB$, so $CF$ (is) to $DF$.

### Modern Formulation

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

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

Proofs: 1

Proofs: 1 2 3 4
Sections: 5

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### 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