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

Scholium. Models of the regular polyeḍrons may be easily obtained as follows: Let the figures represented below be accurately drawn on card-board and cut out entire. At the lines separa

ting two adjacent polygons let the card-board be cut half through; the edges of the several polygons in each figure may then be brought together so as to represent a regular polyedron, and they may be secured in their place by gluing the edges.

To compute the volume of a regular tetraedron.

Let A-BCD be a regular tetraedron; it is required to determine its volume.

From one angle, A, let fall the perpendicular AE upon the opposite face BCD. By Def. 5, the faces of the tetraedron are all equal triangles, therefore AB, AC, AD are equal to each other. Hence they are equally distant from the perpendicular (B. VII., Pr. 5, Cor.); that is, E is the centre of a circle described about the equilateral triangle BCD. The area of the triangle BCD is

B

equal to

BC2
4

√3 (B. VI., Pr. 4, Sch. 2).

Since EF is one half of EC (B. VI., Pr. 4), it is one third of FC or AF. Then, in the triangle AEF, we have (preceding figure) AE2-AF2-FE2-AF2-AF2-AF2.

that is, the volume of a regular tetraedron is equal to the cube of a linear edge multiplied by one twelfth the square root of two.

Cor. The entire surface of the tetraedron is equal to four times the area of the triangle BCD; or BC2√3; that is, the surface of a regular tetraedron is equal to the square of a linear edge multiplied by the square root of thrce.

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 centre.

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

2. A radius of a sphere is a straight line drawn from the centre to any point of the surface. A diameter is any straight line drawn through the centre, 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. 1) that every section of a sphere måde by a plane is a circle. A great circle is a section made by a plane which passes through the centre of the sphere. A small circle is a section made by a plane which does not pass through the centre.

4. The poles of a circle of a sphere are the extremities of that diameter of the sphere which is perpendicular to the plane of

the circle.

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

6. A spherical polygon is a portion of the surface of a sphere bounded by three or more arcs of great circles, each of which is less than a semi-circumference. These arcs are called the sides of the polygon; and the angles which their planes make with each other are the angles of the polygon.

7. A spherical triangle is a spherical polygon of three sides. It is called right-angled, isosceles, or equi lateral in the same cases as a plane triangle.

8. A lune is a portion of the surface of a sphere included between two semi-circumferences of great circles having a common diameter.

9. A spherical ungula or wedge is a portion of a sphere included between the halves of two great circles, and has the lune for its base.

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

11. A zone is a portion of the surface of a sphere included between two parallel planes. 12. A spherical segment is a portion of a sphere included between two parallel planes.

13. The bases of the segment are the sections of the sphere made by the parallel planes; the altitude of the segment or zone is the distance

between the planes. One of the two planes may touch the sphere, in which case the segment has but one base.

14. When a semicircle, revolving about its diameter, describes a sphere, any sector of the semicircle describes a solid, which is called a spherical sector.

Thus, when the semicircle AEB, revolving about its diameter AB, describes a sphere, any circular sector, as ACD or DCE, describes a spherical sector.

E

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

Let ABD be a section made by a plane

in a sphere whose centre is C. From the A point C draw CE perpendicular to the plane ABD; and draw lines CA, CB, CD, etc., to different points of the curve ABD which bounds the section.

The oblique lines CA, CB, CD are equal, because they are radii of the sphere; there

A

B

B

A

B

fore they are equally distant from the perpendicular CE (B. VII., Pr. 5, Cor,). Hence all the lines EA, EB, ED are equal; and, consequently, the section ABD is a circle, of which E is the centre. Therefore every section, etc.

Cor. 1. If the section passes through the centre of the sphere, its radius will be the radius of the sphere; hence all great circles of a sphere are equal to each other.

Cor. 2. Any two great circles of a sphere bisect each other; for, since they have the same centre, their common section is a diameter of both, and therefore bisects both.

Cor. 3. Every great circle divides the sphere and its surface into two equal parts. For if the two parts are separated and applied to each other, base to base, with their convexities turned the same way, the two surfaces must coincide; otherwise there would be points in these surfaces unequally distant from the

centre.

Cor. 4. The centre of a small circle and that of the sphere are in a straight line perpendicular to the plane of the small circle.

Cor. 5. The circle which is farthest from the centre is the least; for the greater the distance CE, the less is the chord AB, which is the diameter of the small circle ABD.

Cor. 6. An arc of a great circle may be made to pass through any two points on the surface of a sphere; for the two given points, together with the centre of the sphere, make three points which are necessary to determine the position of a plane. If, however, the two given points were situated at the extremities of a diameter, these two points and the centre would then be in one straight line, and any number of great circles might be made to pass through them.

A plane perpendicular to a diameter at its extremity touches the sphere.

Let ADB be a plane perpendicular to the diameter DC at its. extremity D, then the plane ADB touches the sphere at the point D.

Let E be any other point in the plane ADB, and join DE, CE. Because CD is perpendicular to the plane ADB, it is perpendicu