Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume

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Theorem

In the words of Euclid:

Parallelepidedal solids which are on the same base and of the same height, and in which the extremities of the sides which stand up are on the same straight lines, are equal to one another.

(The Elements: Book $\text{XI}$: Proposition $29$)


Proof

Euclid-XI-29.png

Let $CM$ and $CN$ be parallelepipeds on the same bases and of the same height.

Let the endpoints of their vertical sides:

$AG, AF, LM, LN, CD, CE, BH, BK$

be on the same straight lines $FN$ and $DK$.

It is to be demonstrated that the parallelepiped $CM$ is equal to the parallelepiped $CN$.


We have that each of the figures $CH$ and $CK$ is a parallelogram.

Thus from Proposition $34$ of Book $\text{I} $: Opposite Sides and Angles of Parallelogram are Equal:

$CB$ equals each of the straight lines $DH$ and $EK$.

Thus $DH = EK$.

Let $EH$ be subtracted from both $DH$ and $EK$.

Therefore their remainders $DE$ and $HK$ are equal.

So from:

Proposition $8$ of Book $\text{I} $: Triangle Side-Side-Side Congruence

and:

Proposition $4$ of Book $\text{I} $: Triangle Side-Angle-Side Congruence

it follows that:

$\triangle DCE = \triangle HBK$

and from Proposition $36$ of Book $\text{I} $: Parallelograms with Equal Base and Same Height have Equal Area:

the parallelogram $DG$ equals the parallelogram $HN$.

For the same reason:

$\triangle AFG = \triangle MLN$

But the parallelogram $CF$ equals the parallelogram $BM$, as they are opposite.

Therefore:

the prism contained by $\triangle AFG$ and $\triangle DCE$ and the three parallelograms $AD, DG, CG$

equals:

the prism contained by $\triangle MLN$ and $\triangle HBK$ and the three parallelograms $BM, HN, BN$.

Let there be added to each the solid of which the parallelogram $AB$ is its base and $GEHM$ its opposite.

Therefore the whole parallelepiped $CM$ is equal to the whole parallelepiped $CN$.

$\blacksquare$


Historical Note

This proof is Proposition $29$ of Book $\text{XI}$ of Euclid's The Elements.


Sources

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