(iv) The regular polyhedron of which each solid angle is formed by three squares is called a cube. It has six faces, eight vertices, (v) The polyhedron of which each solid angle is formed by three regular pentagons is called a regular dodecahedron. It has twelve faces, twenty vertices, thirty edges. THEOREM 7. If F denote the number of faces, E of edges, and V of vertices in any polyhedron, then will E+2=F+V. Suppose the polyhedron to be formed by fitting together the faces in succession: suppose also that Er denotes the number of edges, and V, of vertices, when r faces have been placed in position, and that the polyhedron has n faces when complete. Now when one face is taken there are as many vertices as edges, that is, E1 =V1. The second face on being adjusted has two vertices and one edge in common with the first; therefore by adding the second face we increase the number of edges by one more than the number of vertices; .. E2-V2 = 1. Again, the third face on adjustment has three vertices and two edges in common with the former two faces; therefore on adding the third face we once more increase the number of edges by one more than the number of vertices; Similarly, when all the faces but one have been placed in position, En-1-Vn-1=n-2. But in fitting on the last face we add no new edges nor vertices; :. E=En-1, V=Vn-1 and F= =n. So that E-V=F -2, or, E+2=F+V. This is known as Euler's Theorem. MISCELLANEOUS EXAMPLES ON SOLID GEOMETRY. 1. The projections of parallel straight lines on any plane are parallel. 2. If ab and cd are the projections of two parallel straight lines AB, CD on any plane, shew that AB : CD=abcd. 3. Draw two parallel planes one through each of two straight lines which do not intersect and are not parallel. 4. If two straight lines do not intersect and are not parallel, on what planes will their projections be parallel? 5. Find the locus of the middle point of a straight line of constant length whose extremities lie one on each of two non-intersecting straight lines. MISCELLANEOUS EXAMPLES ON SOLID GEOMETRY. 455 6. Three points A, B, C are taken one on each of the conterminous edges of a cube: prove that the angles of the triangle ABC are all acute. 7. If a parallelepiped is cut by a plane which intersects two pairs of opposite faces, the common sections form a parallelogram. 8. The square on the diagonal of a rectangular parallelepiped is equal to the sum of the squares on the three edges conterminous with the diagonal. 9. The square on the diagonal of a cube is three times the square on one of its edges. 10. The sum of the squares on the four diagonals of a parallelepiped is equal to the sum of the squares on the twelve edges. 11. If a perpendicular is drawn from a vertex of a regular tetrahedron on its base, shew that the foot of the perpendicular will divide each median of the base in the ratio 2: 1. 12. Prove that the perpendicular from the vertex of a regular tetrahedron upon the opposite face is three times that dropped from its foot upon any of the other faces. 13. If AP is the perpendicular drawn from the vertex of a regular tetrahedron upon the opposite face, shew that 3AP2=2a2, where a is the length of an edge of the tetrahedron. 14. The straight lines which join the middle points of opposite edges of a tetrahedron are concurrent. 15. If a tetrahedron is cut by any plane parallel to two opposite edges, the section will be a parallelogram. 16. Prove that the shortest distance between two opposite edges of a regular tetrahedron is one half of the diagonal of the square on an edge. 17. In a tetrahedron if two pairs of opposite edges are at right angles, then the third pair will also be at right angles. 18. In a tetrahedron whose opposite edges are at right angles in pairs, the four perpendiculars drawn from the vertices to the opposite faces and the three shortest distances between opposite edges are concurrent. 19. In a tetrahedron whose opposite edges are at right angles, the sum of the squares on each pair of opposite edges is the same. 20. The sum of the squares on the edges of any tetrahedron is four times the sum of the squares on the straight lines which join the middle points of opposite edges. 21. In any tetrahedron the plane which bisects a dihedral angle divides the opposite edge into segments which are proportional to the areas of the faces meeting at that edge. 22. If the angles at one vertex of a tetrahedron are all right angles, and the opposite face is equilateral, shew that the sum of the perpendiculars dropped from any point in this face upon the other three faces is constant. 23. Shew that the polygons formed by cutting a prism by parallel planes are equal. 24. Three straight lines in space OA, OB, OC, are mutually at right angles, and their lengths are a, b, c: express the area of the triangle ABC in its simplest form. 25. Find the diagonal of a regular octahedron in terms of one of its edges. 26. Shew how to cut a cube by a plane so that the lines of section may form a regular hexagon. 27. Shew that every section of a sphere by a plane is a circle. 28. Find in terms of the length of an edge the radius of a sphere inscribed in a regular tetrahedron. 29. Find the locus of points in a given plane at which a straight line of fixed length and position subtends a right angle. 30. A fixed point O is joined to any point P in a given plane which does not contain O; on OP a point Q is taken such that the rectangle OP, OQ is constant: shew that Q lies on a fixed sphere. GLASGOW PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACLEHOSE AND CC. LTD. WORKS BY H. S. HALL, M.A., and F. H. STEVENS, M.A. A SCHOOL GEOMETRY, based on the recommendations of the Mathematical Association, and on the recent report of the Cambridge Syndicate on Geometry. Parts I. and II.—Part I. Lines and Angles. Part II. Areas of Rectilineal Figures. stance of Euclid Book I. Crown Svo. Rectilineal Figures. Containing the sub1s. 6d. Part III.--Circles. Containing the substance of Euclid Book III. Geometrical equivalents of Containing the substance of 35-37. 6d. 1s. 6d. Part V.-Containing the substance of Euclid Book VI. Parts I.-V. in one volume. 4s. 6d. 18. 6d. 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