THEOREM XXV.

735. In a sphere the sum of one pair of opposite angles of a quadrilateral inscribed in a circle equals the sum of the other pair.

E

C

B

A

Join E, the circumcenter, with A, B, C, D, the vertices of the inscribed quadrilateral.

By 696, ABC = X BAE + BCE, and X ADC =

+ X DCE,

:. ABC + × ADC = 4 BAD + 4 BCD.

¥

X DAE

arcs.

THEOREM XXVI.

736. In equal circles, equal angles at the pole stand on equal

THEOREM XXVII.

737. Equal spherical chords cut equal circles into the same

738. In equal circles, angles at the corresponding poles have the same ratio as their arcs.

THEOREM XXIX.

739. Four pairs of equal circles can be drawn to touch three non-concurrent lines in a sphere.

BOOK X.

POLYHEDRONS.

740. A Polyhedron is a solid bounded by planes.

741. The bounding planes, by their intersections, determine the Faces of the polyhedron, which are polygons.

742. The Edges of a polyhedron are the sects in which its. faces meet.

743. The Summits of a polyhedron are the points in which its edges meet.

744. A Plane Section of a polyhedron is the polygon in which a plane passing through it cuts its faces.

745. A Pyramid is a polyhedron of which all the faces, except one, meet in a point.

746. The point of meeting is called the Apex, and the face not passing through the apex is taken as the Base.

747. The faces and edges which meet at the apex are called Lateral Faces and Edges.

748. Two polygons are said to be parallel when each side of the one is parallel to a corresponding side of the other.

749. A Prism is a polyhedron two of whose faces are congruent parallel polygons, and the other faces are parallelograms.

750. The Bases of a prism are the congruent parallel polygons.

751. The Lateral Faces of a prism are all except its bases. 752. The Lateral Edges are the intersections of the lateral faces.

753. A Right Section of a prism is a section by a plane perpendicular to its lateral edges.