as their altitudes; and pyramids generally are to each other as the products of their bases by their altitudes.

Cor. 3. Similar pyramids are to each other as the cubes of their homologous edges.

Scholium. The solidity of any polyedron may be found by dividing it into pyramids, by planes passing through its

vertices.

PROPOSITION XVIII.

THEOREM.

A frustum of a pyramid is equivalent to the sum of three pyramids, having the same altitude as the frustum, and whose bases are the lower base of the frustum, its upper base, and a mean proportional between them.

Case first. When the base of the frustum is a triangle. Let ABC-DEF be a frustum of a triangular pyramid. If a plane be made to pass through the points A, C, E, it will cut off the pyramid E-ABC, whose altitude is the altitude of the frustum, and its base is ABC, the lower base of the frustum.

Pass another plane through the points C, D, E; it will cut off the pyramid C-DEF, whose altitude is that of the frustum, and its base is DEF, the upper base of the frustum.

C

To find the magnitude of the remaining pyramid E-ACD, draw EG parallel to AD; join CG, DG. Then, because the two triangles AGC, DEF have the angles at A and D equal to each other, we have (Prop. XXIII., B. IV.)

AGC: DEF:: AGXAC: DEXDF,

:: AC: DF, because AG is equal to DE.

Also (Prop. VI., Cor. 1, B. IV.),

ACB: ACG:: AB: AG or DE.

But, because the triangles ABC, DEF are similar (Prop. XIII.), we have

AB: DE: AC: DF.

Therefore (Prop. IV., B. II.),

ACB: ACG:: ACG: DEF;

that is, the triangle ACG is a mean proportional between ACB and DEF, the two bases of the frustum.

Now the pyramid E-ACD is equivalent to the pyramid G-ACD, because it has the same base and the same altitude; for EG is parallel to AD, and, consequently, parallel to the G

plane ACD. But the pyramid G-ACD has the same altitude as the frustum, and its base ACG is a mean proportional be tween the two bases of the frustum.

Case second. When the base of the frustum is any polygon. Let BCDEF-bcdef be a frustum of any pyramid.

康康

Let G-HIK be a triangular pyramid having the same altitude and an equivalent base with the pyramid A-BCDEF, and from it let a frustum HIK-hik be cut B off, having the same altitude with the frustum BCDEFbcdef. The entire pyramids are equivalent (Prop. XVII.) and the small pyramids A-bcdef, G-hik are also equivalent, for their altitudes are equal, and their bases are equivalent (Prop. XIII., Cor. 2). Hence the two frustums are equivalent, and they have the same altitude, with equivalent bases. But the frustum HIK-hik has been proved to be equivalent to the sum of three pyramids, each having the same altitude as the frustum, and whose bases are the lower base of the frustum, its upper base, and a mean proportional between them. Hence the same must be true of the frustum of any pyramid Therefore, a frustum of a pyramid, &c.

PROPOSITION XIX. THEOREM.

There can be but five regular polyedrons.

Since the faces of a regular polyedron are regular poly gons, they must consist of equilateral triangles, of squares, of regular pentagons, or polygons of a greater number of sides.

First. If the faces are equilateral triangles, each solid angle of the polyedron may be contained by three of these tri

angles, forming the tetraedron; or by four, forming the oc taedron; or by five, forming the icosaedron.

No other regular polyedron can be formed with equilat. eral triangles; for six angles of these triangles amount to

four right angles, and can not form a solid angle (Prop. XVIII., B. VII.).

Secondly. If the faces are squares, their angles may be united three and three, forming the hexaedron, or cube.

Four angles of squares amount to four right angles, and can not form a solid angle.

Thirdly. If the faces are regular pentagons, their angles may be united three and three, forming the regular dodecaedron. Four angles of a regular pentagon, are greater than four right angles, and can not form a solid angle.

Fourthly. A regular polyedron can not be

formed with regular hexagons, for three angles of a regular hexagon amount to four right angles. Three angles of a regular heptagon amount to more than four right angles; and the same is true of any polygon having a greater number of sides.

Hence there can be but five regular polyedrons; three formed with equilateral triangles, one with squares, and one with pentagons.

BOOK IX.

SPHERICAL GEOMETRY

Definitions.

1. A sphere is a solid bounded by a curved surface, all the points of which are equally distant from a point within, called the center.

The sphere may be conceived to be described by the revolution of a semicircle ADB, about its diameter AB, which remains unmoved.

2. The radius of a sphere, is a straight line drawn from the center to any point of the surface. The diameter, or axis, is a line passing through the center, and terminated. each way by the surface.

All the radii of a sphere are equal; all the diameters are also equal, and each double of the radius.

3. It will be shown (Prop. I.), that every section of a sphere made by a plane is a circle. A great circle is a section made by a plane which passes through the center of the sphere. Any other section made by a plane is called a smal circle.

4. A plane touches a sphere, when it meets the sphere, but, being produced, does not cut it.

5. The pole of a circle of a sphere, is a point in the surface equally distant from every point in the circumference of this circle. It will be shown (Prop. V.), that every circle, whether great or small, has two poles.

6. A spherical triangle is a part of the surface of a sphere, bounded by three arcs of great circles, each of which is less than a semicircumference. These arcs are called the sides of the triangle; and the angles which their planes make with each other, are the angles of the triangle.

7. A spherical triangle is called right-angled, isosceles, or equilateral, in the same cases as a plane triangle.

8. A spherical polygon is a part of the surface of a sphere bounded by several arcs of great circles.

9. A lune is a part of the surface of a sphere included between the halves of two great circles.

10. A spherical wedge, or ungula, is that portion of the sphere included between the same semicircles, and has the lune for its base.

11. A spherical pyramid is a portion of the sphere included between the planes of a solid angle, whose vertex is at the center. The base of the pyramid is the spherical polygon intercepted by those planes.

12. A zone is a part of the surface of a sphere included between two parallel planes. 13. A spherical segment is a portion of the sphere included between two parallel planes. 14. The bases of the segment are the sections of the sphere; the altitude of the segment, or zone, is the distance between the

sections. One of the two planes may touch the sphere, in which case the segment has but one base. 15. A spherical sector is a solid described by the revolution of a circular sector, in the same manner as the sphere is described by the revolution of a semicircle.

While the semicircle ADB, revolving round its diameter AB, describes a sphere, every circular sector, as ACE or ECD, describes a spherical sector.

16. Two angles which are together equal to two right angles; or two arcs

D

E

B

which are together equal to a semicircumference, are called the supplements of each other.