159. THEOREM. Two triangular pyramids having a triedral angle of the one equal to a triedral angle of the other, are to each other as the products of the edges including the equal triedral angles.

Place the equal triedral angles in coincidence at O.

CP and C'P' perpendicular to the face OA'B'.

Draw

SECTION IV.

THE SPHERE.

DEFINITIONS.

160. A sphere is a solid bounded by a curved surface,

every point of which is equally distant from a fixed point within. The fixed

P

point is the centre. Any line from the centre to the surface is a radius. Any line through the centre limited by the surface is a diameter.

161. A plane or a line is tangent to a sphere when it touches the surface of the sphere in only one point.

From the above definitions the student may readily deduce the following:

162. Radii of the same or equal spheres are equal.

163. Diameters of the same or equal spheres are equal.

164.

Two spheres of equal radii may be made to coincide by placing their centres in coincidence.

165. A sphere may be generated by the revolution of a semicircle about the diameter as an axis.

166. Sections of a sphere through the centre are equal circles of the same radius as the sphere. Such circles are called great circles.

167. Any great circle bisects the sphere.

168. Two great circles bisect each other.

169. A great circle is determined by two points in the surface of the sphere, unless the points are extremities of a

diameter. 170. The plane perpendicular to the radius of a sphere at its extremity is tangent to the sphere. And conversely the plane which is tangent to a sphere is perpendicular to the radius drawn to the point of contact.

171. The distance between two points on the surface of a sphere is measured on the arc of a great circle joining the points.

172. A sphere is inscribed in a polyedron when the faces of the polyedron are tangent to the sphere. The polyedron is then circumscribed about the sphere.

173.

is a circle.

PROPOSITION XXVI.

THEOREM. Every section of a sphere made by a plane

B

The section through the centre is a circle (166). Let ABC be any section not through the centre O; then will ABC be a circle.

Draw a diameter perpendicular to the section ABC, piercing it at some point O'. From the centre O to any points B and A in the perimeter of the section ABC, draw OB and OA. Draw BO' and AO'.

Now, in the triangles OBO' and OAO', the angle 00'A equals the angle OO'B (4); OB equals AO being radii, and OO' is common; hence the triangles are equal, and therefore That is, any points B and A in the perim

AO' equals BO'.

eter of the section ABC are equally distant from O', a point within. By definition then, ABC is a circle having O' for its centre.

Q. E. D.

174. SCHOLIUM. A diameter of a sphere perpendicular to a circle is termed the axis of the circle. The extremities of the axis are the poles of the circle.

175. COROLLARY. The axis of any circle of a sphere pierces the centre of the circle.

For the diameter perpendicular to any circle ABC has been shown to pierce the centre of ABC.

PROPOSITION XXVII.

176. THEOREM. Either pole of a circle of a sphere is equally distant from any two points in the circumference of the circle.

B

P'

Let A and B be any two points in the circumference of the circle ABC whose poles are P and P'; then will the arcs AP and BP be equal, also the arcs AP' and BP' will be equal.

Let O be the intersection of the axis PP' with the circle ABC. Draw the straight lines AO, BO, AP, and BP.

Now, in the triangles AOP and BOP, the angle AOP equals the angle BOP (4); the side AO equals the side BO (175); and OP is common to the two triangles; hence, the triangles are equal and homologous parts AP and BP are equal. Then the arc AP equals the arc BP (374). In like manner we may prove that the arc AP' equals the arc BP'.

Q. E. D.

177.

SCHOLIUM. The distance from the pole of a circle of a sphere to a point in its circumference is called the polar distance of the circle.

178. COROLLARY 1. The polar distances of two equal circles of the same or equal spheres are equal.

For, the diameters of equal circles of a sphere are chords of equal arcs of great circles, each of these equal arcs being twice the polar distance of one of these equal circles.

B

P

C

179. COROLLARY 2. The polar distance of a great circle is a quadrant.

For, AOP is a right angle (174); hence arc AP which is the measure of AOP is a quadrant. In like manner arc BP is found to be a quadrant.

180. COROLLARY 3. That point on the surface of a sphere which is a quadrant's distance from two points in the circumference of a great circle is the pole of the great circle.

PROPOSITION XXVIII.

181. THEOREM. The surface of a sphere is equal to the area of four great circles.

Let the sphere whose centre is D be generated by the revolution of the semicircle FBL about the diameter FL as an axis; and let R represent the radius and S the surface