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