BOOK VIII. POLYHEDRAL ANGLES

AND THE SPHERE

SECTION I. DEFINITIONS; SECANTS AND TANGENTS

187. The Sphere. The geometric solid bounded by a surface all points of which are equidistant from a point within, called the center, is a sphere. The bounding surface is called a spherical surface.

188. Generation of a Sphere. A sphere can be generated by the rotation of a semicircle about its diameter as an axis, for all points on the surface of the solid thus generated will be at a distance from the center of the semicircle equal to a radius.

189. Radii and Diameters. A line from the center of a sphere to its surface is a radius, while a line through the center, joining two points on the surface, is a diameter.

*190. All radii of a sphere are equal. All diameters of a sphere are equal.

*191. Spheres having equal radii are congruent and conversely.

For, if superposed with centers coinciding (that is, placed so that they are concentric), no point on one surface could be off the other surface. (Why?) Note also that equivalent spheres must have equal radii, for if not, the one having the longer radius could contain the other.

From this it follows that only the word equal need be used for spheres, for with them there is no distinction between equivalence and congruence. Compare with the circle.

*192. The locus of points at a given distance from a given point is the surface of a sphere having the given point as its center, and the given distance as its radius.

208. Find the locus of the vertex of the right angle of a right triangle, having a given fixed hypotenuse.

193. Relative Positions of a Point and a Sphere. A point is within a sphere if its distance from the center is less than the length of the radius; on the surface, if its distance from the center is equal to the length of the radius; outside, if its distance from the center is greater than the length of the radius. This follows from the definition of a sphere, and the ordinary meanings of "within" and "outside."

* 194. One, and but one, spherical surface can be drawn through four non-coplanar points. For there is one and but one point equidistant from the four points. See § 91.

195. Inscribed Polyhedrons. A sphere is said to be circumscribed about a polyhedron if its surface contains the vertices of the polyhedron; the polyhedron is said to be inscribed in the sphere. §194 might have been worded: One and but one sphere can be circumscribed about a given tetrahedron.

196. Relative Positions of a Line and a Sphere. A line can lie wholly outside a sphere, for the sphere is limited in size, or it can have one or more points in common with the sphere. If the distance of the line from the center is equal to a radius, it has but one point in common with the sphere (why?), and so is called tangent to the

sphere, the point being called the point of contact. If the distance of the line from the center is less than a radius, two, and but two, lines of length equal to a radius can be drawn to it from the center (determining the plane of the line and the center and using plane geometry); so the line meets the surface in two points and is called a secant. Show that a line tangent to a sphere is not tangent to all the circles through its point of contact.

209. A line whose distance from the center of a sphere is more than the length of a radius does not meet the surface of the sphere.

210. A line tangent to a sphere must be perpendicular to the radius drawn to its point of contact.

211. The plane perpendicular to a tangent at its point of contact passes through the center of the sphere.

197. Relative Positions of a Plane and a Sphere. A plane is outside a sphere if its distance from its center is more than the length of a radius. Why?

A plane is tangent to a sphere if it has one point in common with the sphere. The possibility of such a plane is shown in § 198. The common point is called, as in the case of the tangent line, the point of contact.

A plane is a secant plane if it cuts the sphere, the intersection with the surface necessarily (§ 101) being a closed line. See § 203.

* 198. A plane perpendicular to a radius of a sphere at its surface extremity is tangent to the sphere.

This proves that there can be a plane tangent to a sphere at any point on its surface. Why?

*199. A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact.

For the point of tangency is nearest the center. This proves that there can be but one tangent plane at a point on the surface. Why? §§ 198 and 199 show that a tangent plane is a plane whose distance from the center equals the length of the radius. Similarly a secant plane is one whose distance from the center is less than the length of the radius.

*200 If a plane is tangent to a sphere, a perpendicular to it at the point of contact passes through the center of the sphere.

212. All lines tangent to a sphere at a point are perpendicular to the radius drawn to that point.

213. All lines tangent to a sphere at a point are in the plane tangent to the sphere at that point.

214. Two lines tangent to a sphere at a point determine the plane tangent to the sphere at that point.

201. Circumscribed Polyhedrons. A polyhedron is said to be circumscribed about a sphere, and the sphere to be inscribed in the polyhedron, if the sphere is tangent to all the faces of the polyhedron.

*202. One, and but one, sphere can be inscribed in any given tetrahedron. For there is one point equi

distant from its faces.

203. Theorem I. Every plane section of a sphere is a circle whose center is the foot of the perpendicular from the center of the sphere to its plane.

Plane sections of a sphere are called circles of the sphere. "Circle" is usually used for "circle of a sphere."

215. If a secant plane is gradually moved farther from the center of a sphere, describe the change in its intersection with the surface.

216. A line tangent to a sphere is tangent to every circle through the point of contact, in whose plane it lies.

The diameter perpen

204. Axis and Poles of a Circle. dicular to a circle is called its axis; the points where the axis cuts the surface of the sphere are called the poles of the circle.

205. COR. 1. A line perpendicular to a circle at its center, or one joining the center of the circle to the center of the sphere (unless they are the same point), is the axis of the circle.

206. COR. 2. Sections through the center of a sphere are all equal, and are the largest circles of the sphere. Of others, two that are equidistant from the center are equal, and conversely; and, of two not equidistant from the center, the nearer is the greater, and conversely.

207. Great and Small Circles.

A section through the

center of a sphere is called a great circle; any other sec

tion is called a small circle. In the diagram, G is a great circle, and s is a small circle. The axis of s is PP', its poles being P and P'. What relation to one another have r (the radius of the sphere), r' (the radius of the small circle), and d (its distance from the center)?