The circles of latitude and the meridians on the earth's surface are familiar examples of circumferences of circles of a sphere. The equator and the meridians are circumferences of great circles.

*208. The center of a great circle is the center of the sphere.

*209. Any two great circles of a sphere intersect in a diameter.

*210. A great circle bisects the sphere and its surface.

*211. Two points on the surface (not extremities of a diameter) and the center of the sphere determine a great circle; any three points on the surface determine a circle.

212. COR. 3. Parallel circles of a sphere have the same axis and the same poles.

The circles of latitude on the earth's surface have the axis of the earth as their axis, and its poles as their poles.

213. COR. 4. A great circle through the poles of another circle is perpendicular to it; a great circle perpendicular to another circle contains its axis and its poles. This is sometimes stated: For a great circle to be perpendicular to another circle, it is necessary, and it is sufficient, that it contain its poles.

217. A great circle that contains one of the poles of another circle must contain the other pole also.

214. COR. 5. The locus of points in a sphere equidistant from all points on the circumference of a circle is the axis of the circle; the poles of a circle are equidistant from all points on its circumference.

215. Polar Chords and Polar Distances. The chords from the pole of a circle to points on its circumference are called polar chords; the great circle ares from its pole to points on its circumference are called polar distances. Unless otherwise stated the chords and distances from the nearer pole are meant.

*216. The polar distances of the same, or of equal, circles on a sphere are equal.

217. A Quadrant. One quarter of the circumference of a great circle is called a quadrant of the sphere. The degree measure of a quadrant is 90°.

* 218. The polar distance of a great circle is a quadrant, and conversely.

Use the central angles subtended by the polar distances. *219. If two points on a spherical surface are a quadrant's distance apart on the great circle arc through them, each is the pole of a great circle through the other.

Using either as a pole, draw the axis, then draw the great circle of which it is the pole.

*220. If a point on the surface of a sphere is at a quadrant's distance from each of two other points of the surface, it is the pole of a great circle through them. Proceed as in § 218 or § 219. What cases are there? 221. Angles between Arcs. The angle between two arcs is the angle between the tangent lines at the point of intersection.

218. Two coplanar circles are perpendicular to each other when a radius of one is tangent to the other. Must a radius of each arc be tangent to the other circle?

222. Spherical Angles. The most important class of

angles between arcs is that having great circle arcs as the arms of the angle. Such an angle is called a spherical angle. As two great circles intersect in a diameter, the tangents are both perpendicular to this diameter, and therefore are the arms of a measuring angle of the dihedral angle between the great circles.

*223. The angle between two great circle arcs equals the measuring angle of the dihedral angle between their planes.

*224. A great circle arc through a pole of another great circle is perpendicular to its circumference.

225. Measurement of Spherical Angles. The most convenient way to measure a spherical angle is by the measuring angle of the corresponding dihedral angle, drawn

T

B

at the center of the sphere. This angle is evidently in the great circle perpendicular to the edge of the dihedral angle, and therefore perpendicular to the planes in which the arms of the spherical angle lie.

The spherical angle XTY equals ATB or XOY, measuring angles of the dihedral angle between the planes TXT and TYT. But for might be measured by arc XY; therefore it follows that

*226. A spherical angle is measured by the subtended great circle arc having its vertex as a pole.

219. Explain how a figure can be drawn so that it will be bounded by three great circle arcs, each perpendicular to the others.

THE MATERIAL SPHERE

*227. To find the radius of a given circumference on a given material sphere.

Select three points on the circumference, and consider a triangle, with these three points as vertices, as inscribed in the circle. Construct, in some plane, the triangle having these three sides (using an ordinary compass to carry the lengths, for it will measure a chord of a circle on a sphere just as well as a sect on a plane). Circumscribe a circle about this triangle, and it will equal the given circle. Why?

*228. To describe a circumference on a sphere with a given pole, and a given polar chord.

*229. The polar chord of a circle of a sphere is the mean between the diameter of the sphere and its own projection upon the axis of the circle; the radius of a circle is the mean between the two sects it makes on the axis of that circle.

Pass a great circle through the pole, and work in that plane.

230. Theorem II. To find the diameter of a given material sphere.

With any pole, and any polar chord that is of convenient length, describe a circumference on the sphere; find the radius of this circle, and from these two lengths, construct the right triangle considered in § 229.

220. On a sphere of known diameter, to construct a circumference such that a triangle inscribed in it shall be congruent to a given triangle.

221. If the diameter of a material sphere is known, to construct on it a circumference of given radius.

222. To construct a circumference on a given material sphere so that it will be a given distance from the center.

223. To construct parallel circumferences on a material sphere.

224. Given any circumference on a sphere, to find any number of points on the circumference of the great circle perpendicular to the given circle at any given point.

231. Relative Position of Two Spheres. Two spheres either lie entirely outside each other; lie outside each other except for one point in common; have more than one point in common, without either being entirely contained in the other; or one is entirely contained in the other. Only the second and third of these possibilities are of special interest in solid geometry.

232. Center Line. The line through the centers of two spheres is called their line of centers. The sect between the centers is called their center sect.

233. Tangent Spheres. Two spheres whose surfaces have one point in common are called tangent spheres, the point of tangency being called the point of contact. They are said to be externally or internally tangent, according as they are outside each other, or one is contained in the other. Unless otherwise stated, tangent will be used to mean externally tangent.

*234. Two spheres whose surfaces meet in the center line are tangent.

*235. Two spheres tangent to the same plane at the same point are tangent.