these bodies thus: first, draw these figures on card-board; then cut them out, and also cut half through the board along the dotted lines. We can then bring the faces together in the required shape.

1. The polyedron which has for its vertices the centres of the four faces of a regular tetraedron is also a regular tetraedron.

2. The polyedron which has for its vertices the centres of the six faces of a regular tetraedron, is a regular octaedron.

3. Conversely, the polyedron which has for vertices the centres of the eight faces of a regular octaedron, is a regular hexaedron.

4. The polyedron which has for its vertices the centres of the faces of a regular dodecaedron, is a regular icosaedron.

5. Conversely, the polyedron which has for vertices the centres of the faces of a regular icosaedron, is a regular dodecaedron.

6. The polyedron which has for its vertices the middle points of the six edges of a regular tetraedron, is a regular octaedron.

7. Show that the volume of the regular tetraedron formed as indicated in 1, is of the volume of the first tetraedron.

PROBLEMS.

1. Compute the volume of a regular octaedron whose edge is given. 2. Compute the volume of a regular octaedron, the vertices of which are the middle points of the six edges of a regular tetraedron whose edge is given.

3. Compute the volume of a regular octaedron, the vertices of which are the centres of the six faces of a regular hexaedron whose edge is given.

4. Compute the volume of a regular hexaedron, whose vertices are the centres of the eight faces of a regular octaedron whose edge is given.

BOOK VII.

THE SPHERE.

DEFINITIONS.

1. The sphere is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the centre. The sphere may be conceived as generated by the revolution of a semicircle, DAE, about its diameter, DE; for the surface described in this movement

D

F

G

N

2. The radius of a sphere is a straight line drawn from the centre to any point of the surface; the diameter or axis is a line passing through the centre and terminated on both sides by the surface.

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

3. All the sections of a sphere by planes passing through its centre are obviously equal circles, which have the centre for their common centre, and for radii the radii of the sphere. It will be shown (Prop. II.), that all other sections of a sphere by planes, will be circles of smaller radii than the radii of the

sphere. This granted, a great circle is a section which passes through the centre; a small circle one which does not pass through the centre.

4. A plane is tangent to a sphere when their surfaces have but one point in common. This point is called the point of contact or of tangency.

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

6. The angle of two arcs of great circles on the sphere, is the angle of the planes of those arcs. Thus, the angle ADM (Fig. Def. 1), of the arcs AD and MD, is the angle of the planes ACD and MCD.

7. A spherical triangle is a portion of the surface of a sphere bounded by three arcs of great circles.

These arcs, named the sides of the triangle, are always supposed to be each less than a semi-circumference. ABC (Fig. Def. 12) is a spherical triangle of which AB, AC, and BC, are the sides.

8. A spherical triangle takes the name of scalene, isosceles, equilateral, in the same cases as a rectilineal triangle.

9. A spherical polygon is a portion of the surface of a sphere terminated by several arcs of great circles. MN (Fig. Def. 12) is a spherical polygon. The spherical triangle is the simplest of the spherical polygons.

10. A lune is that portion of the surface of a sphere which is included between two great semi-circumferences meeting in the extremities of a common diameter,

U

с

B

12. A spherical pyramid is a portion of the solid sphere included between the planes of a solid angle whose vertex is the centre, and which terminate in the surface of the sphere. c The base of the pyramid is the spherical polygon intercepted by these planes.

B

13. A zone is the portion of the surface of the sphere included between two parallel planes which form its bases. Ι One of these planes may be tangent to the sphere; in which case the zone has B only a single base.

B'

Ꭺ

14. A spherical segment is the portion of the solid sphere included between two parallel planes which form its bases. One of these planes may be tangent to the sphere; in which case the segment has only a single base. The segment has a zone for the E curved or spherical part of its surface.

15. The altitude of a zone or of a segment is the distance between

the two parallel planes which form the bases of the zone or segment, or the distance from the point of contact of the parallel tangent plane to the base, when they have only a single base.

16. Whilst the semicircle, DAE (Fig. Def. 1), revolving round its diameter, DE, describes the sphere, any circular sector, as DCF or FCH, describes a solid which is named a spherical sector.

17. A polyedron is said to be inscribed in a sphere when the vertices of all its solid angles are on the surface of the sphere.

18. A polyedron is said to be circumscribed about a sphere when all its faces are tangent planes to the spherical surface.

PROPOSITION I.

THEOREM.

A straight line cannot meet the surface of a sphere in more than two points.

The demonstration is exactly the same as that of Proposition III., Book II.

PROPOSITION II.

THEOREM.

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

Let AMB be a section made by a plane in the sphere whose centre is C. From the point C, draw CO, perpendicular to the plane A AMB; and different lines, CM, CM, to different points of the curve AMB, which terminates the section.

D

M

B

M

C

The oblique lines CM, CM, CB are equal, since they are radii of the sphere; they are therefore equally distant from CO (Book V., Prop. IX.); hence all the lines OM, OM, OB are equal; hence the section AMB is a circle, of which the point O is the centre.

COR. 1. If the section passes through the centre of the sphere, its radius will be the radius of the sphere; and all great circles are equal, as we have seen before (Def. 3). All other sections, since they have