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BOOK VII.

POLYHEDRO JS, CYLINDERS, AND CONES.

GENERAL DEFINITIONS. 493. DEF. A Polyhedron is a solid bounded by four or more polygons.

A polyhedron bounded by four polygons is called a tetrahedron; by six, a hexahedron; by eight, an octahedron; by twelve, a dodecahedron; by twenty, an icosahedron.

494. DEF. The Faces of a polyhedron are the bounding polygons.

495. DEF. The Edges of a polyhedron are the intersections of its faces.

496. DEF. The Vertices of a polyhedron are the intersections of its edges.

497. Def. A Diagonal of a polyhedron is a straight line joining any two vertices not in the same face.

498. DEF. A Section of a polyhedron is a polygon formed by the intersection of a plane with three or more faces.

499. DEF. A Convex polyhedron is a polyhedron every section of which is a convex polygon.

500. DEF. The Volume of a polyhedron is the numerical measure of its magnitude referred to some other polyhedron as a unit of measure.

501. Def. The polyhedron adopted as the unit of measure is called the Unit of Volume.

502. DEF. Similar polyhedrons are polyhedrons which have the same form.

503. DEF. Equivalent polyhedrons are polyhedrons which have the same volume.

504. DEF. Equal polyhedrons are polyhedrons which have the same form and volume.

On Prisms. 505. DEF. A Prism is a polyhedron two of whose faces are equal and parallel polygons, and the other faces are parallelograms.

OBLIQUE PRISM.

506. DEF. The Bases of a prism are the equal and parallel polygons.

507. DEF. The Lateral faces of a prism are all the faces except the bases.

508. DEF. The Lateral or Convex Surface of a prism is the sum of its lateral faces.

509. DEF. The Lateral edges of a prism are the intersections of its lateral faces; the Basal edges of a prism are the intersections of the bases with the lateral faces.

510. DEF. Prisms are triangular, quadrangular, pentangular, etc., according as their bases are triangles, quadrangles, pentagons, etc.

511. DEF. A Right prism is a prism whose lateral edges are perpendicular to its bases.

512. DEF. An Oblique prism is a prism whose lateral edges are oblique to its bases.

513. DEF. A Regular prism is a right prism whose bases are regular polygons, and hence its lateral faces are equal rectangles.

514. DEF. The Altitude of a prism is the perpendicular distance between the planes of its bases. The altitude of a right prism is equal to any one of its lateral edges.

515. DEF. A Truncated prism is a portion of a prism included between either base and a section inclined to the base and cutting all the lateral edges.

516. DEF. A Right section of a prism is a section perpendicular to its lateral edges.

517. DEF. A Parallelopiped is a prism whose bases are parallelograms.

518. DEF. A Right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases; hence its lateral faces are rectangles.

519. DEF. An Oblique parallelopiped is a parallelopiped whose lateral edges are oblique to its bases.

520. DEF. A Rectangular parallelopiped is a right parallelopiped whose bases are rectangles.

521. DEF. A Cube is a rectangular parallelopiped all of whose faces are squares.

RIGHT PRISM,

PROPOSITION I. THEOREM. 522. The sections of a prism made by parallel planes are equal polygons.

Let the prism A D be intersected by the parallel

planes GK, G' K'.
We are to prove section G H IKL = section G' H' I' K'L'.

G H, HI, I K, etc., are parallel respectively to G' H', H' I', I' K', etc.,

$ 465 (the intersections of two Il planes by a third plane are Il lines).

..GHI, H I K, etc., are equal respectively to [ G' H'I', H' I' K', etc.,

§ 462 (two & not in the same plane, having their sides respectively parallel and

lying in the same direction, are equal). Also, sides GH, HI, IK, etc., are equal respectively to G' H', H'I', I' K', etc.,

§ 135 (II lines comprehended between II lines are equal). .. section G H IKL= section G' H'I' K'I', $ 155 (being mutually equiangular and equilateral).

Q. E. D.

523. COROLLARY. Any section of a prism parallel to the base is equal to the base ; and all right sections of a prism are equal.

PROPOSITION II. THEOREM. . 524. The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section.

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Let G H IKL be a right section of the prism A D'.

We are to prove lateral area of prism A D' = A A X perimeter G H I K L.

. Consider the lateral edges A A, B B', etc., to be the bases of the S A B', B C', etc., which form the convex surface of the

prism.

Then the altitudes of these s will be the L G H, HI, IK, etc.,

and the area of each o is the product of its base and altitude.

§ 321 Now the bases of these 5 are all equal, § 464

(Il lines comprehended between Il planes are equal); . and the sum of the altitudes G H, HI, I K, etc., is the perimeter of the right section.

Hence, the sum of the areas of these 5 is the product of a lateral edge A A' by the perimeter of the right section.

That is, the lateral area of the prism is equal to the product of a lateral edge by the perimeter of a right section.

Q. E. D.

525. COROLLARY. The lateral area of a right prism is equal to the altitude multiplied by the perimeter of the base.

PROPOSITION III. THEOREM. 526. Two prisms are equal if the three faces including a trihedral angle of the one be respectively equal to the three corresponding faces including a trihedral angle of the other, and similarly placed.

BIC
Let A D, AG, AJ, be respectively equal to A'D', A' ,

A' J', and similarly placed.
We are to prove prism A I = prism A' I'.

Now trihedral Z A = trihedral Z A', $ 492 (two trihedrals are equal, when the three face és of the one are equal respectively to the three face of the other and are similarly placed).

Apply trihedral Z A to trihedral Z A'.
Then the base A D will coincide with the base A'D',

face A G with A' G',

and face A J with A'I';
.. F G will coincide with F'G', and FJ with F" J'.

.. the upper bases, FI and F I', will coincide,
(being equal polygons, since they are equal to the equal lower bases).

.. the remaining edges will coincide,

(their extremitics being the same points).
.. the prisms will coincide and be equal...

Q. E. D. 527. COROLLARY 1. Two truncated prisms are equal, if the three faces including a trihedral of the one be respectively equal to the three faces including a trihedral of the other, and be similarly placed.

528. CoR. 2. Two right prisms having equal bases and altitudes are equal. If the faces be not similarly placed, if one be inverted, the faces will be similarly placed and the prisms can be made to coincide.

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