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655. Book IX. will develop the Geometry of the Sphere, from theorems and problems almost identical with those whose assumption gave us Plane Geometry. In Book VIII., these have been demonstrated by considering the sphere as contained in ordinary tri-dimensional space. But, if we really confine ourselves to the sphere itself, they do not admit of demonstration, except by making some more difficult assumption : and so they are the most fundamental properties of this surface and its characteristic line, the great circle; just as the assumptions in our first book were the most fundamental properties of the plane and its characteristic line.

So now we will call a great circle simply the spherical line ; and, whenever in this book the word line is used, it means spherical line. Sect now means a part of a line less than a half-line.

Two-Dimensional Definition of the Sphere.

656. Suppose a closed line, such that any portion of it may be moved about through every portion of it without any other change. Suppose a portion of this line is such, that, when moved on the line until its first end point comes to the trace of its second end point, that second end point will have moved to the trace of its first end point. Call such a portion a halfline, and any lesser portion a sect. Suppose, that, while the extremities of a half-line are kept fixed, the whole line can be so moved that the slightest motion takes it completely out of its trace, except in the two fixed points. Such motion would generate a surface which we will call the Sphere.

Fundamental Properties of the Sphere.


657. A figure may be moved about in a sphere without any other change; that is, figures are independent of their place on the sphere.

658. Through any two points in the sphere can be passed a line congruent with the generating line of the sphere. In Book IX., the word line will always mean such a line, and sect will mean a portion of it less than half.

659. Two sects cannot meet twice on the sphere ; that is, if two sects have two points in common, the two sects coincide between those two points.

660. If two lines have a common sect, they coincide throughout. Therefore through two points, not end points of a half-line, only one distinct line can pass.

661. A sect is the smallest path between its end points in the sphere.

662. A piece of the sphere from along one side of a line will fit either side of any other portion of the line.


663. If one end point of a sect is kept fixed, the other end point moving in the sphere describes what is called an arc, while the sect describes at the fixed point what is called a spherical angle. The angle and arc are greater as the amount of turning in the sphere is greater.

664. When a sect has turned sufficiently to fall again into the same line, but on the other side of the fixed point or vertex, the angle described is called a straight angle, and the arc described is called a semicircle.

665. Half a straight angle is called a right angle.

666. The whole angle about a point in a sphere is called a perigon: the whole arc is called a circle.

The point is called the pole of the circle, and the equal sects are called its spherical radii.


667. A circle can be described from any pole, with any sect as spherical radius.

668. All straight angles are equal.

669. COROLLARY I. All perigons are equal.

670. COROLLARY II. If one extremity of a sect is in a line, the two angles on the same side of the line as the sect are together a straight angle.

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671. COROLLARY III. Defining adjacent angles as two angles having a common vertex, a common arm, and not overlapping, it follows, that, if two adjacent angles together equal a straight angle, their two exterior arms fall into the same line.

672. COROLLARY IV. If two sects cut one another, the vertical angles are equal.

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673. Any line turning in the sphere about one of its points, through a straight angle, comes to coincidence with its trace, and has described the sphere.

674. COROLLARY I. The sphere is a closed surface.

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