PROPOSITION IV. THEOREM.

529. An oblique prism is equivalent to a right prism whose bases are equal to right sections of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism.

Let A D' be an oblique prism, and FI a right section.

Complete the right prism FI', making its edges equal to

those of the oblique prism.

We are to prove oblique prism A D'≈ right prism F I'.

In the prisms AI and A' I'

trihedral A trihedral

A',

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(two trihedrals are equal when three face of the one are respectively equal to three face of the other, and are similarly placed).

(two truncated prisms are equal when the three faces including a trihedral

of the one are respectively equal to the three faces including a trihedral

of the other, and are similarly placed).

To each of these equal prisms add the prism F D'.
Then oblique prism A D' right prism FI'.

Q. E. D.

We are to prove faces A F and DG equal and parallel.

(two not in the same plane having their sides || and lying in the same

direction are equal).

(if two not in the same plane have their sides || and lying in the same direction their planes are parallel).

allel.

In like manner we may prove A H and BG equal and par

Q. E. D.

531. SCHOLIUM. Any two opposite faces of a parallelopiped may be taken for bases, since they are equal and parallel parallelograms.

PROPOSITION VI. THEOREM.

532. The plane passed through two diagonally opposite edges of a parallelopiped divides the parallelopiped into two equivalent triangular prisms.

G

H

E

L

D

K

J

B

Let the plane A EGC pass through the opposite edges A E and C G of the parallelopiped A G.

We are to prove that the parallelopiped AG is divided into two equivalent triangular prisms, ABC-F, and AD C-H.

Let IJKL be a right section of the parallelopiped made by a plane to the edge A E.

The intersection IK of this plane with the plane AEG C is the diagonal of the

IJK L.

.. AIK J=A I KL.

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But prism ABC-F is equivalent to a right prism whose base is IJK and whose altitude is A E,

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(any oblique prism is to a right prism whose bases are equal to right sections of the oblique prism, and whose altitude is equal to a lateral edge of the oblique prism).

The prism A D C-H is equivalent to a right prism whose base is ILK, and whose altitude is A E.

Now the two right prisms are equal,
(two right prisms having equal bases and altitudes are equal).

.. ABC-FA DC-H.

$529

$ 528

Q. E. D

533. Two rectangular parallelopipeds having equal bases are to each other as their altitudes.

Let AB and A'B' be the altitudes of the two rectangular parallelopipeds, P, and P', having equal bases.

We are to prove

CASE I.

When A B and A'B' are commensurable.

Find a common measure m, of A B and A' B'.

Suppose m to be contained in AB 5 times, and in A' B' 3 times.

At the several points of division on A B and A'B' pass planes to these lines.

The parallelopiped P will be divided into 5,

and P' into 3, parallelopipeds equal, each to each, § 528 (two right prisms having equal bases and altitudes are equal).

Let A B be divided into any number of equal parts,

and let one of these parts be applied to A'B' as many times as A'B' will contain it.

Since A B and A' B' are incommensurable, a certain number of these parts will extend from A' to a point D, leaving a remainder D B' less than one of these parts.

Through D pass a plane to A' B', and denote the parallelopiped whose base is the same as that of P', and whose altitude is A' D by Q.

Now, since A B and A' D are commensurable,

Q: P

=

A'DA B.

(Case I.)

Suppose the number of parts into which A B is divided to be continually increased, the length of each part will become less and less, and the point D will approach nearer and nearer to B'.

The limit of Q will be P',

and the limit of A' D will be A'B',

.. the limit of Q: P will be P': P,

and the limit of A' D: A B will be A'B': A B,

Moreover the corresponding values of the two variables Q : P and A'D: A B are always equal, however near these variables approach their limits.

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.. their limits P': P

A'B': A B.

Q. E. D.

534. SCHOLIUM. The three edges of a rectangular parallelopiped which meet at a common vertex are its dimensions. Hence two rectangular parallelopipeds which have two dimensions in common are to each other as their third dimensions.