---
book: 5
number: 19
id: "V.19"
kind: "theorem"
uses: ["[[book-5/proposition-16]]", "[[book-5/proposition-17]]", "[[book-5/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.19"
license: "CC-BY-SA-4.0"
---

# V.19

*If, as a whole is to a whole, so is a part subtracted to a part subtracted*, *the remainder will also be to the remainder as whole to whole*.

## Proof

For, as the whole *AB* is to the whole *CD*, so let the part *AE* subtracted be to the part *CF* subtracted; I say that the remainder *EB* will also be to the remainder *FD* as the whole *AB* to the whole *CD*.

For since, as *AB* is to *CD*, so is *AE* to *CF*, alternately also, as *BA* is to *AE*, so is *DC* to *CF*. [[book-5/proposition-16|V. 16]]

And, since the magnitudes are proportional *componendo*, they will also be proportional *separando*, [[book-5/proposition-17|V. 17]] that is, as *BE* is to *EA*, so is *DF* to *CF*, and, alternately, as *BE* is to *DF*, so is *EA* to *FC*. [[book-5/proposition-16|V. 16]]

But, as *AE* is to *CF*, so by hypothesis is the whole *AB* to the whole *CD*.

Therefore also the remainder *EB* will be to the remainder *FD* as the whole *AB* is to the whole *CD*. [[book-5/proposition-11|V. 11]]

Therefore etc. [

Porism. From this it is manifest that, if magnitudes be proportional componendo, they will also be proportional convertendo.

] Q. E. D.