G.531 parallelepipedal solid cut — Heath

XI.25

If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.

For let the parallelepipedal solid < A B C D > be cut by the plane < F G > which is parallel to the opposite planes < R A >, < D H >; I say that, as the base < A E F V > is to the base < E H C F >, so is the solid < A B F U > to the solid < E G C D >.

For let / A H / be produced in each direction, let any number of straight lines whatever, / A K /, / K L /, be made equal to / A E /, and any number whatever, / H M /, / M N /, equal to / E H /; and let the parallelograms < L P >, < K V >, < H W >, < M S > and the solids < L Q >, < K R >, < D M >, < M T > be completed.

Then, since the straight lines / L K /, / K A /, / A E / are equal to one another,

the parallelograms < L P >, < K V >, < A F > are also equal to one another, < K O >, < K B >, < A G > are equal to one another, and further < L X >, < K Q >, < A R > are equal to one another, for they are opposite. [G.530]

For the same reason the parallelograms < E C >, < H W >, < M S > are also equal to one another, < H G >, < H I >, < I N > are equal to one another, and further < D H >, < M Y >, < N T > are equal to one another.

Therefore in the solids < L Q >, < K R >, < A U > three planes are equal to three planes.

But the three planes are equal to the three opposite;

therefore the three solids < L Q >, < K R >, < A U > are equal to one another.

For the same reason the three solids < E D >, < D M >, < M T > are also equal to one another.

Therefore, whatever multiple the base < L F > is of the base < A F >, the same multiple also is the solid < L U > of the solid < A U >.

For the same reason, whatever multiple the base < N F > is of the base < F H >, the same multiple also is the solid < N U > of the solid < H U >.

And, if the base < L F > is equal to the base < N F >, the solid < L U > is also equal to the solid < N U >; if the base < L F > exceeds the base < N F >, the solid < L U > also exceeds the solid < N U >; and, if one falls short, the other falls short.

Therefore, there being four magnitudes, the two bases < A F >, < F H >, and the two solids < A U >, < U H >, equimultiples have been taken of the base < A F > and the solid < A U >, namely the base < L F > and the solid < L U >, and equimultiples of the base < H F > and the solid < H U >, namely the base < N F > and the solid < N U >, and it has been proved that, if the base < L F > exceeds the base < F N >, the solid < L U > also exceeds the solid < N U >, if the bases are equal, the solids are equal, and if the base falls short, the solid falls short.

Therefore, as the base < A F > is to the base < F H >, so is the solid < A U > to the solid < U H >. [G.176] Q. E. D.

Dependency Graph

Required for

G.541, G.542, G.543, G.544, G.546, G.547, G.550, G.554, G.555, G.556, G.557, G.558, G.559, G.560, G.563, G.565, G.570, G.571, G.572, G.573, G.577, G.578