---
book: 5
number: 24
id: "V.24"
kind: "theorem"
uses: ["[[book-5/proposition-22]]", "[[book-5/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.24"
license: "CC-BY-SA-4.0"
---

# V.24

*If a first magnitude have to a second the same ratio as a third has to a fourth*, *and also a fifth have to the second the same ratio as a sixth to the fourth*, *the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth*.

## Proof

Let a first magnitude *AB* have to a second *C* the same ratio as a third *DE* has to a fourth *F*; and let also a fifth *BG* have to the second *C* the same ratio as a sixth *EH* has to the fourth *F*; I say that the first and fifth added together, *AG*, will have to the second *C* the same ratio as the third and sixth, *DH*, has to the fourth *F*.

For since, as *BG* is to *C*, so is *EH* to *F*, inversely, as *C* is to *BG*, so is *F* to *EH*.

Since, then, as *AB* is to *C*, so is *DE* to *F*, and, as *C* is to *BG*, so is *F* to *EH*, therefore, ex aequali, as *AB* is to *BG*, so is *DE* to *EH*. [[book-5/proposition-22|V. 22]]

And, since the magnitudes are proportional separando, they will also be proportional componendo; [[book-5/proposition-18|V. 18]] therefore, as *AG* is to *GB*, so is *DH* to *HE*.

But also, as *BG* is to *C*, so is *EH* to *F*; therefore, ex aequali, as *AG* is to *C*, so is *DH* to *F*. [[book-5/proposition-22|V. 22]]

Therefore etc. Q. E. D.