VOLUME OF SPHERE.

It has been proved that the quadrilaterals CDEF, cdef are equal in

area.

The volume of the pyramid OCDEF is equal to OP. CDEF, because OP is perpendicular to CDEF.

And the volume of the prism McfNde is equal to half the orthohedron, whose base is cdef and height Lp, that is, equal to

OP. CDEF.

Hence the volume of OCDEF is two-thirds of the volume of

McfNde.

Again, if lunes be drawn as before and between the planes forming each lune a pyramid and a prism be constructed corresponding to OCDEF, McfNde respectively and then the number of the lunes be increased indefinitely, the volume generated by the rotation of the triangle OQR round AOB is two-thirds of the volume of the cylinder generated by the rotation of qrNM round AOB.

Again, if a polygon of an even number of sides be described in the plane APB as before and a strip of a cone generated by the rotation of each of its sides, except those which touch at A and B, the volume generated by the rotation of the polygon omitting the part formed by the two triangles, whose vertices are at O and whose bases are the tangents which touch at A and B, is two-thirds of the volume of the cylinder.

Again, if the number of sides of the polygon be increased indefinitely, the volume generated by that part of the polygon becomes equal to the volume of the sphere.

Hence the volume of the sphere is two-thirds of the volume of the enveloping cylinder.

Similarly it may be proved that, if two closed curves be drawn on the sphere and on the cylinder so that the straight line joining corresponding points on the two curves always cuts the axis of the cylinder at right angles, the volume of the portion of the sphere enclosed by radii of the sphere passing through the curve on the sphere is twothirds of the volume of the portion of the cylinder enclosed by straight lines drawn at right angles to the axis passing through the curve on the cylinder.

The relations which we have established between the surfaces and the volumes of the sphere and the enveloping cylinder were discovered by Archimedes, a Greek philosopher, who was killed at the siege of Syracuse, 212 B.C.

MISCELLANEOUS EXERCISES.

1. If a straight line be parallel to each of two planes it is parallel to their common section.

2. The perpendicular drawn from a vertex of a regular tetrahedron on the opposite face is three times that drawn from its own foot on any of the other faces.

3. A pyramid stands on an equilateral triangle as base and the angles at the vertex are right angles: shew that the sum of the perpendiculars on the faces from any point of the base is constant.

4. The edges AB, CD of a tetrahedron are at right angles, and K, K' are the orthocentres of the faces ABC, ABD. Prove that KK', AB are at right angles.

5. If OABC be a tetrahedron, prove geometrically that it is impossible for the opposite edges OA, BC, and OB, CA and OC, AB to be inclined to one another at the same angle unless that angle be a right angle. The order in which the letters are given is to be considered as fixing which angle between each pair of lines is considered.

6. If ABCD be an orthocentric tetrahedron and O its orthocentre, then the five tetrahedrons formed by any four of the five points O, A, B, C, D is an orthocentric tetrahedron and the fifth point is the orthocentre.

7. If ABCD be an orthocentric tetrahedron and O its orthocentre, the twelve-point spheres of the five tetrahedrons formed by joining the points O, A, B, C, D intersect each other in pairs in the nine-point circles of the ten triangles formed by joining the points O, A, B, C, D.

8. In an orthocentric tetrahedron each solid angle is contained by three plane angles of the same kind, i.e. all acute, all right, or all obtuse.

9. In a tetrahedron ABCD, AB is at right angles to CD. Also the sum of the inclinations of the planes meeting in AB, and of the planes meeting in CD is equal to two right angles; and the planes bisecting these angles meet in the shortest distance EF between AB and CD. Prove that the rectangle AB, CD is four times the square on EF.

10. ABCD is a given tetrahedron, A', B', C' points on DA, DB, DC taken so that the planes ABC, A'B'C' are parallel; prove that the locus of the intersection of the planes AB'C', BC'A', CA'B' is the line joining D to the centre of gravity of the triangle ABC.

11. A point moves so that the sum of the volumes of the tetrahedrons formed by joining it to the angular points of two given triangles is constant: prove that its locus is a plane.

12. The section of a rectangular parallelepiped made by a plane passing through one of its edges is a rectangle.

13. The middle points of the six edges of a cube which do not meet a diagonal of the cube lie in a plane which bisects the diagonal and is at right angles to it.

14. Every triangular section of a rectangular parallelepiped is an acute-angled triangle.

15. In a parallelepiped, AB, the diagonal of one face is at right angles to the edges of intersection of the four faces not parallel to the first. A similar fact is true for AC, AD, the diagonals of the other faces meeting at A.

Shew that the parallelepiped is rectangular.

16. The section of a parallelepiped by every plane through the centre of the parallelepiped is either a parallelogram or a hexagon whose opposite sides are equal and parallel.

17. The sum of squares on the diagonals of a parallelepiped is equal to the sum of the squares on the edges.

18. Prove that, if OP be a diagonal of a parallelepiped, and OA, OB, OC three edges, then the sum of the squares on OP, OA, OB, OC is equal to the sum of the squares on PA, PB, PC.

19. An octahedron can be formed of any eight equal triangles provided that they are acute angled.

20. Two intersecting circles cut at the same angle as their inverse circles.

21. Prove that a sphere can be described through any two circles which intersect in two points. What are the conditions that a sphere can be described through two non-intersecting circles?

22. If six points A, B, C, D, E, F be such that three circles ABCD, CDEF, EFAB may be described through them in fours, the six points lie on a sphere. What are the conditions that a sphere can be described through three non-intersecting circles?

23. A straight line is divided at a fixed point into parts of a constant length, and the ratios of the distances of its extremities from two other fixed points are equal: prove that the line always lies in the same plane.

24. Shew how to reduce the Solid problem to describe a sphere to touch two given planes and to pass through two given points to the Plane problem to find a point on a given straight line equidistant from a given point and another given straight line.

25. In a tetrahedron of which the opposite edges are equal, shew that (i) the centre of gravity, the centre of the inscribed sphere and the centre of circumscribing sphere all coincide, (ii) the shortest distance between opposite edges bisects those edges, and (iii) that the three shortest distances are all mutually at right angles and are bisected in the centre of gravity.

26. Each edge of a tetrahedron is equal to the opposite edge: prove that the sum of the squares on the sides of a face is double the square on the diameter of the circumscribed sphere.

27. It is possible to describe two circular cones such that each passes through two given circular sections of the same sphere. The vertices of the two cones lie on the polar line of the common section of the planes of the two circles.

28. Every tetrahedron, self-conjugate* with respect to a sphere, is orthocentric.

29. An orthocentric tetrahedron ABCD and its orthocentre O are such that each of the five tetrahedrons is self-conjugate with respect to a sphere, whose centre is at the fifth point. These five spheres cut each other at right angles. One sphere has an imaginary, and the other four real radii.

30. The six spheres, described on the edges of a self-conjugate tetrahedron as diameters, all cut the sphere at right angles.

31. If two points P, Q be conjugate with respect to a circle in a plane in which they lie, they are conjugate with respect to any sphere passing through the circle.

32. If PQR be a self-conjugate triangle with respect to the circular section of a sphere made by the plane PQR, and if S be the pole of the plane PQR with respect to the sphere, the tetrahedron PQRS is self-conjugate with respect to the sphere.

33. In a self-conjugate tetrahedron one vertex is within the sphere and the other three without. All the plane angles of the tetrahedron at the internal vertex are obtuse angles.

34. If PQRS be a self-conjugate tetrahedron, a cone, whose vertex is R or S drawn through a circle, whose plane passes through PQ, will cut the sphere again in a second circle, whose plane also passes through PQ.

35. Prove that any two, reverse tetrahedrons may be divided each into six parts, which in pairs are equal in all respects, each part being composed of two tetrahedrons and the twelve tetrahedrons having a common vertex at the centre of the inscribed sphere.

36. If a, ẞ, y, d be the centroids of the faces of the tetrahedron ABCD opposite A, B, C, D, and if O be the centre of the circle ABC, H the foot of the perpendicular from D on the plane ABC, and P the orthocentre of the triangle ABC, then the centre of the section of the sphere aßyd by the plane ABC is the centroid of the triangle OHP.

*Two points are said to be conjugate with respect to a circle or a sphere, when each point is on the polar line or the polar plane of the other point; a triangle or a tetrahedron is said to be self-conjugate with respect to a circle or a sphere when each vertex is on the polar line or the polar plane of each of the other vertices.