BOOK SEVENTH.

DEFINITIONS.

1. A prism is a solid contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others are parallelograms. To construct this solid, let ABCDE be any rectilineal figure. In a plane parallel to ABC, draw

the lines FG, GH, HI, &c. parallel to the sides AB, BC, CD, &c.; thus there will be formed a figure FGHIK, similar to ABCDE. Now let the vertices of the corresponding angles be joined by the lines AF, BG, CH, &c.; the faces ABGF, BCHG, &c. will evidently be parallelograms, and the solid thus formed will be a prism.

B

K

2. The equal and parallel plane figures ABCDE, FGHIK are called the bases of the prism. The other planes or parallelograms, taken together, constitute the lateral or convex surface of the prism.

3. The altitude of a prism is the perpendicular distance between its bases; and its length is a line equal to any one of its lateral edges, as AF or BG, &c.

4. A right prism is one in which the lateral edges AF, BG, &c. are perpendicular to the planes of its bases; then each of them is equal to the altitude of the prism: in every other case, the prism is oblique.

5. A prism is triangular, quadrangular, pentagonal, etc., according as the base is a triangle, à quadrilateral, a pentagon, etc.

6. A prism which has a parallelogram for its base, has all its faces parallelograms, and is called a pa- G rallelopipedon. A parallelopipedon is rectangular, when all its faces are rectangles.

H

F

B

7. When the faces of a rectangular parallelopipedon are square, it is called a cube.

8. A pyramid is a solid formed by several triangular planes which meet in a point, as S, and terminate in the same plane rectilineal figure ABCDE.

D

B

A

The plane figure ABCDE is called the base of the pyramid; the point S is its vertex ; and the triangles ASB, BSC, &c. taken together, form the convex or lateral surface of the pyramid.

9. The altitude of a pyramid is the perpendicular drawn from the vertex to the plane of its base, produced if ne

cessary.

10. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, a quadrangle, etc..

11. A pyramid is regular, when its base is a regular figure, and the perpendicular from its vertex passes through

the centre of its base; that is, through the centre of a circle which may be conceived to circumscribe its base.

12. Two solids are similar, when they are contained by the same number of similar planes, similarly situated, and having like inclinations to one another.

PROPOSITION I.

THEOREM. Two prisms are equal, when a solid angle in each is contained by three planes, which are equal in both, and similarly situated.

E

H

b

Let the base ABCDE be equal to the base abcde; the parallelogram ABGF equal to the parallelogram abgf, and the parallelogram BCHG equal to the parallelogram bchg then will the prism ABCI be equal to the prism abci.

For, apply the base ABCDE upon its equal abcde, so that the bases (being equal) may coincide. But the three plane angles which form the solid angle B are respectively equal to the three plane angles which form the solid angle

b; that is, ABC

=

abc, ABG

=

abg, and GBC = gbc, and they are also similarly situated; therefore the solid angles B and C are equal (B. VI, Prop. xx1), and therefore BG will coincide with its equal bg. And it is likewise evident, because the parallelograms ABGF and abgf are equal, that the side GF will coincide with its equal gf, and in the same manner GH with gh; therefore the upper base FGHIK will coincide with its equal fghik, and the two solids will be identical, since their vertices are the

same.

Cor. Two right prisms, which have equal bases and equal altitudes, are equal. For, since the side AB is equal to ab, and the altitude BG to bg, the rectangle ABGF will be equal to abgf, and, in the same way, the rectangle BGHC will be equal to bghc; and thus the three planes which form the solid angle B will be equal to the three planes which form the solid angle b: hence the two prisms are equal.

PROPOSITION II.

THEOREM. In every parallelopipedon, the opposite planes are equal and parallel.

By the definition of this solid, the bases ABCD, EFGH are equal parallelograms, and their sides are parallel it remains only to show that the same is true of any two opposite lateral faces, such as

G

H

C

D

F

A

B

AEHD, BFGC. Now AD is equal and parallel to BC, because the figure ABCD is a parallelogram; for a like reason, AE is parallel to BF: hence the angle DAE is equal to the angle CBF, and the planes DAE, CBF are parallel; hence also the parallelogram DAEH is equal to the parallelogram CBFG. In the same way, it may be shown that the opposite parallelograms ABFE, DCGH are equal and parallel.

Cor. Since the parallelopipedon is a solid bounded by six planes, whereof those lying opposite to each other are equal and parallel, it follows that any face and the one opposite to it may be assumed as the bases of the parallelopipedon.

Scholium. If three straight lines AB, AE, AD, passing through the same point A, and making given angles with each other, are known, a parallelopipedon may be formed on those lines. For this purpose, a plane must be extended through the extremity of each line, and parallel to the plane of the other two; that is, through the point B a plane parallel to DAE, through D a plane parallel to BAE, and through E a plane parallel to BAD. The mutual intersections of these planes will form the parallelopipedon required.