---
book: 5
number: 10
id: "V.10"
kind: "theorem"
uses: ["[[book-5/proposition-7]]", "[[book-5/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.10"
license: "CC-BY-SA-4.0"
---

# V.10

*Of magnitudes which have a ratio to the same*, *that which has a greater ratio is greater*; *and that to which the same has a greater ratio is less*.

## Proof

For let *A* have to *C* a greater ratio than *B* has to *C*; I say that *A* is greater than *B*.

For, if not, *A* is either equal to *B* or less.

Now *A* is not equal to *B*; for in that case each of the magnitudes *A*, *B* would have had the same ratio to *C*; [[book-5/proposition-7|V. 7]] but they have not; therefore *A* is not equal to *B*.

Nor again is *A* less than *B*; for in that case *A* would have had to *C* a less ratio than *B* has to *C*; [[book-5/proposition-8|V. 8]] but it has not; therefore *A* is not less than *B*.

But it was proved not to be equal either; therefore *A* is greater than *B*.

Again, let *C* have to *B* a greater ratio than *C* has to *A*; I say that *B* is less than *A*.

For, if not, it is either equal or greater.

Now *B* is not equal to *A*; for in that case *C* would have had the same ratio to each of the magnitudes *A*, *B*; [[book-5/proposition-7|V. 7]] but it has not; therefore *A* is not equal to *B*.

Nor again is *B* greater than *A*; for in that case *C* would have had to *B* a less ratio than it has to *A*; [[book-5/proposition-8|V. 8]] but it has not; therefore *B* is not greater than *A*.

But it was proved that it is not equal either; therefore *B* is less than *A*.

Therefore etc. Q. E. D.