---
book: 12
number: 18
id: "XII.18"
kind: "theorem"
uses: ["[[book-12/proposition-17]]", "[[book-5/proposition-16]]", "[[book-12/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:12.prop.18"
license: "CC-BY-SA-4.0"
---

# XII.18

*Spheres are to one another in the triplicate ratio of their respective diameters.*

## Proof

Let the spheres *ABC*, *DEF* be conceived, and let *BC*, *EF* be their diameters; I say that the sphere *ABC* has to the sphere *DEF* the ratio triplicate of that which *BC* has to *EF*.

For, if the sphere *ABC* has not to the sphere *DEF* the ratio triplicate of that which *BC* has to *EF*, then the sphere *ABC* will have either to some less sphere than the sphere *DEF*, or to a greater, the ratio triplicate of that which *BC* has to *EF*.

First, let it have that ratio to a less sphere *GHK*, let *DEF* be conceived about the same centre with *GHK*, let there be inscribed in the greater sphere *DEF* a polyhedral solid which does not touch the lesser sphere *GHK* at its surface, [[book-12/proposition-17|XII. 17]] and let there also be inscribed in the sphere *ABC* a polyhedral solid similar to the polyhedral solid in the sphere *DEF*; therefore the polyhedral solid in *ABC* has to the polyhedral solid in *DEF* the ratio triplicate of that which *BC* has to *EF*. [[book-12/proposition-17|XII. 17, Por.]]

But the sphere *ABC* also has to the sphere *GHK* the ratio triplicate of that which *BC* has to *EF*; therefore, as the sphere *ABC* is to the sphere *GHK*, so is the polyhedral solid in the sphere *ABC* to the polyhedral solid in the sphere *DEF*; and, alternately, as the sphere *ABC* is to the polyhedron in it, so is the sphere *GHK* to the polyhedral solid in the sphere *DEF*. [[book-5/proposition-16|V. 16]]

But the sphere *ABC* is greater than the polyhedron in it; therefore the sphere *GHK* is also greater than the polyhedron in the sphere *DEF*.

But it is also less, for it is enclosed by it.

Therefore the sphere *ABC* has not to a less sphere than the sphere *DEF* the ratio triplicate of that which the diameter *BC* has to *EF*.

Similarly we can prove that neither has the sphere *DEF* to a less sphere than the sphere *ABC* the ratio triplicate of that which *EF* has to *BC*.

I say next that neither has the sphere *ABC* to any greater sphere than the sphere *DEF* the ratio triplicate of that which *BC* has to *EF*.

For, if possible, let it have that ratio to a greater, *LMN*; therefore, inversely, the sphere *LMN* has to the sphere *ABC* the ratio triplicate of that which the diameter *EF* has to the diameter *BC*.

But, inasmuch as *LMN* is greater than *DEF*, therefore, as the sphere *LMN* is to the sphere *ABC*, so is the sphere *DEF* to some less sphere than the sphere *ABC*, as was before proved. [[book-12/proposition-2|XII. 2, Lemma]]

Therefore the sphere *DEF* also has to some less sphere than the sphere *ABC* the ratio triplicate of that which *EF* has to *BC*: which was proved impossible.

Therefore the sphere *ABC* has not to any sphere greater than the sphere *DEF* the ratio triplicate of that which *BC* has to *EF*.

But it was proved that neither has it that ratio to a less sphere.

Therefore the sphere *ABC* has to the sphere *DEF* the ratio triplicate of that which *BC* has to *EF*. Q. E. D.