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 A B C is parallel to the plane D E F (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,

SOSP::AB: DE.

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

ABC:DEF::AB: DE.

Multiplying together the corresponding terms of these two proportions, we have

ABCSO: DEFX SP : : A B3 : D E3.

Now, ABCX SO 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 five, 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 C B.

C

D

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

surface equally distant from every point in the circumference of the circle.

502. It will be shown (Prop. V.) that every circle, great or small, has two poles.

503. A PLANE is TANGENT to a sphere, when it meets the sphere in but one point, however far it may be produced.

504. A SPHERICAL ANGLE

is the difference in the direction of two arcs of great circles of the sphere; as A ED, formed by the arcs EA, DE.

It is the same as the angle resulting from passing two planes through those arcs; as the angle formed on the edge EF, by the planes EAF, EDF.

A

F

E

B

505. A SPHERICAL TRIANGLE is a portion of the surface of a sphere bounded by three arcs of great circles, each arc being less than a semi-circumference; as AED.

These arcs are named the sides of the triangle; and the angles which their planes form with each other are the angles of the triangle.

506. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a plane triangle.

507. A SPHERICAL POLYGON is a portion of the surface of a sphere bounded by several arcs of great circles.

508. A LUNE is a portion of the surface of a sphere comprehended between semi-circumferences of two great circles; as AIGBDF.

509. A SPHERICAL WEDGE, or UNGULA, is that portion of a sphere comprehended between

G

A

K

is a portion of the sphere cut off by a plane, or comprehended between two parallel planes.

512. The ALTITUDE of a ZONE or of a SPHERICAL SEGMENT is the perpendicular distance between the two parallel planes which comprehend the zone or segment.

In case the zone or segment is a portion of the sphere cut off, one of the planes is a tangent to the sphere.

513. A SPHERICAL SECTOR is a solid described by the revolution of a circular sector, in the same manner as the semicircle of which it is a part, by revolving round its diameter, describes a sphere.

514. A SPHERICAL PYRAMID is a portion of the sphere comprehended between the planes of a polyedral angle whose vertex is the centre.

The base of the pyramid is the spherical polygon intercepted by the same planes.

PROPOSITION I.THEOREM.

515. Every section of a sphere made by a plane is a circle.

Let ABE be a section made by a plane in the sphere whose centre is C. From the centre, C, draw CD perpendicular to the plane ABE; and draw the lines CA, CB, CE, to different points of the curve ABE, which bounds the section.