Therefore the solid
AE is also equal to the solid
CF.
Therefore etc. Q. E. D.
PROPOSITION 32.
Parallelepipedal solids which are of the same height are to one another as their bases.
Let
AB,
CD be parallelepipedal solids of the same height; I say that the parallelepipedal solids
AB,
CD are to one another as their bases, that is, that, as the base
AE is to the base
CF, so is the solid
AB to the solid
CD.
For let
FH equal to
AE be applied to
FG, [
I. 45] and on
FH as base, and with the same height as that of
CD, let the parallelepipedal solid
GK be completed.
Then the solid
AB is equal to the solid
GK; for they are on equal bases
AE,
FH and of the same height. [
XI. 31]
And, since the parallelepipedal solid
CK is cut by the plane
DG which is parallel to opposite planes, therefore, as the base
CF is to the base
FH, so is the solid
CD to the solid
DH. [
XI. 25]
But the base
FH is equal to the base
AE, and the solid
GK to the solid
AB; therefore also, as the base
AE is to the base
CF, so is the solid
AB to the solid
CD.
Therefore etc. Q. E. D.
PROPOSITION 33.
Similar parallelepipedal solids are to one another in the triplicate ratio of their corresponding sides.
Let
AB,
CD be similar parallelepipedal solids, and let
AE be the side corresponding to
CF; I say that the solid
AB has to the solid
CD the ratio triplicate of that which
AE has to
CF.
For let
EK,
EL,
EM be produced in a straight line with
AE,
GE,
HE, let
EK be made equal to
CF,
EL equal to
FN, and further
EM equal to
FR, and let the parallelogram
KL and the solid
KP be completed.
Now, since the two sides
KE,
EL are equal to the two sides
CF,
FN, while the angle
KEL is also equal to the angle
CFN, inasmuch as the angle
AEG is also equal to the angle
CFN because of the similarity of the solids
AB,
CD,
