PROPOSITION X. THEOREM.

538. The volume of a rectangular parallelopiped is equal to the product of its three dimensions, the unit of volume being a cube whose edge is the linear unit.

a

P

Let a, b, and c be the three dimensions of the rectangular parallelopiped P, and let the cube U be the unit of volume.

539. COROLLARY I. Since a cube is a rectangular parallelopiped having its three dimensions equal, the volume of a cube is equal to the third power of its edge.

540. COR. II. The product a × b represents the base when c is the altitude; hence: The volume of a rectangular parallelopiped is equal to the product of its base by its altitude.

541. SCHOLIUM. When the three dimensions of the retangular parallelopiped are each exactly divisible by the linear unit, this proposition is rendered evident by dividing the solid into cubes, each equal to the unit of volume. Thus, if the three edges which meet at a common vertex contain the linear unit 3, 4 and 5 times respectively, planes passed through the several points of division of the edges, and perpendicular to them, will divide the solid into cubes, each equal to the unit of volume; and there will evidently be 3 × 4 × 5 of these cubes.

542. The volume of any parallelopiped is equal to the product of its base by its altitude.

G

H

N

E

A

Let ABCD-F be a parallelopiped having all its faces oblique, and HR its altitude.

We are to prove A B C D-FA B C D × HR.

By making the right section HIJN and completing the parallelopiped H I J N-G L K M we have a right parallelopiped equivalent to, ABC D-F.

$529 (an oblique prism is equivalent to a right prism whose base is a right section of the oblique prism and whose altitude is equal to a lateral edge of the oblique prism).

Through the edge IL make the right section IL PO, and complete the right parallelopiped IL PO-HGQR, and we have a rectangular parallelopiped equivalent to HIJ N-GLK M, § 529 and hence equivalent to A B C D-F.

Now

and

DILG HEFGH,

□ OPQR = (ILGH) = JKMN;
□

$322

§ 530

$530

PROPOSITION XII. THEOREM.

543. The volume of any prism is equal to the product of its base by its altitude.

Let V denote the volume, B the base, and H the altitude of the triangular prism A E C-E'.

Upon the edges AE, EC, EE', construct parallelopiped AECD-E'.

(the plane passed through two diagonally opposite edges of a parallelopiped divides it into two equivalent triangular prisms),

$532

§ 133

§ 542

(the volume of any parallelopiped is equal to the product of its base by its

CASE II. When the base is a polygon of more than three sides. Planes passed through the lateral edge A A', and the several diagonals of the base will divide the given prism into triangular prisms,

which have for a common altitude the altitude of the prism. Hence, the volume of the entire prism is the product of the sum of their bases by the common altitude;

that is the entire base by the altitude of the prism.

Q. E. D.

544. COROLLARY. Prisms having equivalent bases are to each other as their altitudes; prisms having equal altitudes are to each other as their bases; and any two prisms are to each other as the product of their bases and altitudes. Any two prisms having equivalent bases and equal altitudes are equivalent.

PROPOSITION XIII. THEOREM.

545. The four diagonals of a parallelopiped bisect each

Let AG, EC, BH, and FD, be the four diagonals of the parallelopiped A G.

We are to prove these four diagonals bisect each other.

.. its diagonals AG and EC bisect each other in the

point 0.

§ 138 In like manner a plane passed through the opposite and || edges FG and AD will form a AFGD,

whose diagonals A G and FD will bisect each other in the point 0.

$ 138 Also, a plane passed through the opposite and I edges EH and BC will form a□E BCH,

whose diagonals E C and B H will bisect each other in the point O.

.. the four diagonals bisect each other at the point 0.

Q. E. D.

546. COROLLARY. The diagonals of a rectangular parallelopiped are equal.

547. SCHOLIUM. The point O, in which the four diagonals intersect, is called the centre of the parallelopiped; and it is evident that any straight line drawn through the point O and terminated by two opposite faces of the parallelopiped is bisected at that point. Hence O is the centre of symmetry.

ON PYRAMIDS.

548. DEF. A Pyramid is a polyhedron one of whose faces is a polygon, and whose other faces are triangles having a common vertex and the sides of the polygon for bases.

549. DEF. The Base of a pyramid is the face whose sides are the bases of the triangles having a common vertex.

550. DEF. The Lateral faces of a pyramid are all the faces except the base.

551. DEF. The Lateral surface of a pyramid is the sum of its lateral faces.

552. DEF. The Lateral edges of a pyramid are the intersections of its lateral faces.

553. DEF. The Basal edges of a pyramid are the intersections of its base with its lateral faces.

554. DEF. The Vertex of a pyramid is the common vertex of its lateral faces.

555. DEF. The Altitude of a pyramid is the perpendicular distance from its vertex to the plane of its base.

Thus, V-A BCDE is a pyramid; ABCDE is its base; A VB, B V C, etc. are its lateral faces, and their sum is its lateral surface; VA, V B, etc. are its lateral edges; A B, BC, etc.

D

V

its basal edges; V is its vertex; VO is its altitude.

A

B