B

There now remains of the frustum the pyramid A CD-E. Draw EG parallel to AD; join C G and DG. Then, since EG is parallel to AD, it is parallel to the plane A CD (Prop. XI. Bk. VII.); and the pyramid ACD-E is equivalent to the pyramid ACD-G, since they have the same base, ACD, and their vertices, E and G, lie in the same straight line parallel to the common base. But the pyramid ACD-G is the same as the pyramid A G C-D, whose altitude is that of the frustum, and whose base, A G C, as will be proved, is a mean proportional between the bases ABC and DEF.

The two triangles A GC, DEF have the angles A and D equal to each other (Prop. XVI. Bk. VII.); hence we have (Prop. XXVIII. Bk. IV.),

AGC: DEF:: AGX AC: DEX DF;

but since A G is equal to D E,

AGC: DEF:: AC: DF.

We have, also (Prop. VI. Cor., Bk. IV.),

ABC: AGC::AB: A G or D E.

But the similar triangles A B C, DEF give

AB: DE:: AC: DF;

hence (Prop. X. Bk. II.),

ABC: AGC:: AGC: DEF;

that is, the base A G C is a mean proportional between the bases ABC, DEF of the frustum.

Secondly. Let GHIKLMNOPQ be the frustum of a pyramid, whose base is any polygon.

Let ABC-S be

S

D

E

The bases of the two pyramids may be regarded as situated in the same plane, in which case the plane MNOPQ produced will form in the triangular pyramid a section, DEF, at the same distance above the common plane of the bases; and therefore the section D E F will be to the section MNOPQ as the base ABC is to the base GHIKL (Prop. XVI. Cor. 1); and since the bases are equivalent, the sections will be so likewise. Hence, the pyramids MNOPQ-T, DEF-S, having the same altitude and equivalent bases, are equivalent. For the same reason, the entire pyramids GHIKL-T, ABC-S are equivalent; consequently, the frustums GHIKLMNOPQ, ABC-DEF, are equivalent. But the frustum ABC-DEF has been shown to be equivalent to the sum of three pyramids having for their common altitude. the altitude of the frustum, and whose bases are the two bases of the frustum, and a mean proportional between them. Hence the proposition is true of the frustum of any pyramid.

PROPOSITION XXII.-THEOREM.

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

homologous polyedral angles at the vertices are equal (Art. 452); hence the smaller pyramid may be so applied to the larger, that the polyedral angle S shall be common to both.

In that case, the bases ABC, DEF will be parallel; for, since the homologous faces are similar, the angle SDE is equal to SA B, and SEF to SBC; hence the plane ABC is parallel to the plane DEF (Prop. XVI. Bk. VII.). Then let SO be drawn from the vertex S perpendicular to the plane A B C, and let P be the point where this perpendicular meets the plane DEF. From what has already been shown (Prop. XVI.), we shall have SO: SP::SA: SD::AB: DE;

and consequently,

SO: SP::AB: DE.

But the bases ABC, DEF are similar; hence (Prop. XXIX. Bk. IV.),

ABC: DEF:: AB2: DE2. Multiplying together the corresponding terms of these two proportions, we have

3

ABCSO: DEFX SP :: A B3: DE3.

Now, ABCXSO represents the solidity of the pyramid ABC-S, and DEFX SP that of the pyramid DEF-S (Prop. XX.); hence two similar pyramids are to each other as the cubes of their homologous edges.

PROPOSITION XXIII.-THEOREM.

495. There can be no more than five regular polyedrons. For, since regular polyedrons have equal regular polygons for their faces, and all their polyedral angles equal, there can be but few regular polyedrons.

First. If the faces are equilateral triangles, polyedrons may be formed of them, having each polyedral angle contained by three of these triangles, forming a solid bounded by four equal equilateral triangles; or by four, forming a solid bounded by eight equal equilateral triangles; or by fire, forming a solid bounded by twenty equal equilateral triangles. No others can be formed with equilateral triangles. For six of these angles are equal to four right angles, and cannot form a polyedral angle (Prop. XX. Bk. VII.).

Secondly. If the faces are squares, their angles may be arranged by threes, forming a solid bounded by six equal squares. Four angles of a square are equal to four right angles, and cannot form a polyedral angle.

Thirdly. If the faces are regular pentagons, their angles may be arranged by threes, forming a solid bounded by twelve equal and regular pentagons.

We can proceed no farther. Three angles of a regular hexagon are equal to four right angles; three of a heptagon are greater. Hence, there can be formed no more than five regular polyedrons, three with equilateral triangles, one with squares, and one with pentagons.

496. Scholium. The regular polyedron bounded by four equilateral triangles is called a TETRAEDRON; the one bounded by eight is called an OCTAEDRON; the one bounded by twenty is called an ICOSAEDRON. The regular polyedron bounded by six equal squares is called a HEXAEDRON, or CUBE; and the one bounded by twelve equal and regular pentagons is called a DODECAEDRON.

BOOK IX.

THE SPHERE, AND ITS PROPERTIES.

DEFINITIONS.

497. A SPHERE is a solid, or volume, bounded by a curved surface, all points of which are equally distant from a point within, called the centre.

The sphere may be conceived to be formed by the revolution of a semicircle, DAE, about its diameter, DE, which remains fixed.

498. The RADIUS of a sphere is a straight line drawn from the centre to any point in surface, as the line CB.

D

C

B

E

The DIAMETER, or AXIS, of a sphere is a line passing through the centre, and terminated both ways by the surface, as the line D E.

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

499. A CIRCLE, it will be shown, is a section of a sphere. A GREAT CIRCLE of the sphere is a section made by a plane passing through the centre, and having the centre of the sphere for its centre; as the section AB, whose centre is C.

500. A SMALL CIRCLE of the sphere is any section made by a plane not passing through the centre.

501. The POLE of a circle of the sphere is a point in the