8. If the four diagonals of a quadrangular prism be concurrent, it is a parallelopiped. 9. If the slant height of a right cone be equal to the diameter of its base, its total surface is to the surface of the inscribed sphere as 9 : 4. 10. The middle points of two pairs of opposite edges of a triangular pyramid are coplanar, and form a parallelogram. 11. If the four perpendiculars from the vertices on the opposite faces of a pyramid ABCD be concurrent, then 12. Every section of a sphere by a plane is a circle. 13. The locus of the centres of parallel sections is a diameter of the sphere. 14. If any number of lines in space pass through a fixed point, the feet of the perpendiculars on them from another fixed point are homospheric. 15. Extend the property of Ex. 4 to the pyramid. 16. The volume of the ring described by a circle which revolves round a line in its plane is equal to the area of the circle, multiplied by the circumference of the circle described by its centre. 17. Any plane bisecting two opposite edges of a tetrahedron bisects its volume. 18. Planes which bisect the dihedral angles of a tetrahedron meet in a point. 19. Planes which bisect perpendicularly the edges of a tetrahedron meet in a point. 20. The volumes of two triangular pyramids, having a common solid angle, are proportional to the rectangles contained by the edges terminating in that angle. 21. A plane bisecting a dihedral angle of a tetrahedron divides the opposite edge into portions proportional to the areas containing that edge. 22. The volume of a sphere: the volume of the circumscribed cube as π : 6. 23. If h be the height, and ρ the radius of a segment of a sphere, its volume is (h2 + 3ρ2). 24. If h be the perpendicular distance between two parallel planes, which cut a sphere in sections whose radii are ρ1, ρ2, the volume of the frustum is {h2 + 3(ρ12 + ρ22)}. 25. If δ be the distance of a point P from the centre of a sphere whose radius is R, the sum of the squares of the six segments at three rectangular chords passing through P is = 6R2 − 2δ2. 26. The volume of a sphere : the volume of an inscribed cube as π : 2. 27. If O–ABC be a tetrahedron whose angles AOB, BOC, COA are right, the square of the area of the triangle ABC is equal to the sum of the squares of the three other triangular faces. 28. In the same case, if p be the perpendicular from O on the face ABC, 29. If h be the height of an æronaut, and R the radius of the earth, the extent of surface visible = . 30. If the four sides of a gauche quadrilateral touch a sphere, the points of contact are concyclic. NOTES. _____ NOTE A. MODERN THEORY OF PARALLEL LINES.