# Volumes of Spheres are in Triplicate Ratio of Diameters

## Theorem

In the words of Euclid:

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

(*The Elements*: Book $\text{XII}$: Proposition $18$)

## Proof

Let the spheres $ABC$ and $DEF$ be described.

Let their diameters be $BC$ and $EF$.

It is to be demonstrated that:

- $ABC : DEF = BC^3 : EF^3$

Suppose it is not the case that:

- $ABC : DEF = BC^3 : EF^3$

Then:

- $ABC : O = BC^3 : EF^3$

where $O$ is either less than or greater than $DEF$.

Let $O$ be equal to a sphere $GHK$ which is less than $DEF$.

Let $DEF$ be described about $GHK$ with the same center as $GHK$.

From Proposition $17$ of Book $\text{XII} $: Construction of Polyhedron in Outer of Concentric Spheres:

- let a polyhedron $X$ be inscribed inside $DEF$ which does not touch $GHK$.

Let a polyhedron $Y$ be inscribed inside $ABC$ which is similar to the one inscribed inside $DEF$.

- $X : Y = BC^3 : EF^3$

But we have that:

- $ABC : GHK = BC^3 : EF^3$

Therefore:

- $ABC : GHK = X : Y$

So from Proposition $16$ of Book $\text{V} $: Proportional Magnitudes are Proportional Alternately:

- $ABC : X = GHK : Y$

But $ABC > X$, as $ABC$ completely encloses $X$.

Therefore $GHK > Y$.

But we have that $GHK < Y$, because $Y$ completely encloses $GHK$.

From this impossibility it follows that it is not the case that:

- $ABC : O = BC^3 : EF^3$

where $O < DEF$.

Similarly it can be shoen that it is not the case that:

- $DEF : O = EF^3 : BC^3$

where $O < ABC$.

Now it is to be shown that it is not the case that:

- $ABC : O = BC^3 : EF^3$

where $O$ is greater than $DEF$.

Suppose that:

- $ABC : LMN = BC^3 : EF^3$

where $LMN$ is a sphere which is greater than $DEF$.

Then:

- $LMN : ABC = EF^3 : BC^3$

We have that:

- $LMN > DEF$

From Lemma to Proposition $2$ of Book $\text{XII} $: Areas of Circles are as Squares on Diameters:

- $LMN : ABC = DEF : Z$

where $Z$ is some sphere which is less than $ABC$.

Therefore:

- $DEF : Z = EF^3 : BC^3$

where $Z$ is less than $ABC$.

This has been shown to be impossible.

Therefore it is not the case that:

- $ABC : LMN : BC^3 : EF^3$

where $LMN > DEF$.

It follows that:

- $ABC : DEF = BC^3 : EF^3$

$\blacksquare$

## Historical Note

This proof is Proposition $18$ of Book $\text{XII}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{XII}$. Propositions