44. Describe a regular hexagon equal to a given square. 45. Describe a regular hexagon equal to a regular pentagon. 46. Describe a square which shall have the same ratio to a given square which one given straight line has to another. 47. Describe a circle equal to the sum of two given circles. 48. Describe a circle equal to the difference of two given circles. 49. Describe a circle equal to a given quadrant. 50. Describe a circle equal to a given semicircle. 51. If circles be described on the sides of an acuteangled triangle as diameters, any two of them will be together greater than the third. 52. Bisect a circle by a concentric circle. 53. Divide a circle into any given number of equal parts by means of concentric circles. 54. If PS the diameter of a circle be trisected in Q, R, and semicircles be described on PQ, PR on one side of PS, and on OS, RS on the other side of PS, then will the circle be divided into three equal parts having equal perimeters. 55. Divide a circle into any number of equal parts having equal perimeters. 56. If on the bounding radii of a quadrant semicircles be described intersecting one another, then will the area common to the two semicircles be equal to the area intercepted between them and the arc of the quadrant; also the two remaining portions of the semicircles will each be equal to the square on the radius of one of the semicircles. 57. The area of a quadrant is half that of the circle described on its chord; also of the three parts of the circle without the quadrant, the crescent will be equal to the triangular portion of the sector, and the two segments of circles will be together equal to the remaining portion of the sector. 58. The angles of equal sectors are inversely as the squares on their radii. SOLID GEOMETRY. 1. Trisect a cube. 2. AB is drawn i to the plane of the right angle BCD; prove that CD is I to the plane of the angle ABC. 3. If three straight lines are at right angles to one another, then is each straight line 1 to the plane passing through the other two. 4. A perpendicular is drawn from the vertex of a regular tetrahedron on the opposite face, and from its foot a perpendicular is drawn to any one of the other faces; prove that the former line is three times the latter. 5. Find the locus of a point in a plane which is always situated at a given distance from a given point out of that plane. 6. Determine the longest and shortest lines which can be drawn to the circumference of a circle from a given point out of the plane of the circle. 7. From the point of intersection of a straight line with a plane draw a straight line in the given plane making the least possible angle with the given straight line. (This angle is the inclination of the given straight line to the plane.) 8. Determine a plane through a given point perpendicular to a given straight line. 9. If the arms of one angle be ll to the arms of another angle, and the straight line joining their vertices is 1 to each of the arms of one angle, then shall it also be 1 to each of the arms of the other angle. 10. If two parallel straight lines meet a given plane, they shall be inclined to it at the same angle. 11. If from a point A above a plane straight lines AB, AC be drawn meeting it in B and C of which AB is perpendicular to the plane, and AC perpendicular to a straight line DC in that plane, and CB joined ; CB shall be perpendicular to DC. 12. If three straight lines are mutually 1 to one another, then are the planes passing through them I to one another. 13. · Draw a straight line 1 to each of two straight lines not in the same plane. 14. Find the shortest path between two straight lines not in the same plane. 15. In any parallelepiped the sum of the squares on the four diagonals is equal to the sum of the squares on the edges. 16. Find the locus of a point equidistant from three given points. 17. AB, BC are = and I to DE, EF respectively, but are not in the same plane with them; prove that AD is = and il to CF. 18. If two straight lines intersecting one another be respectively perpendicular to two planes intersecting one another, then will the common section of these planes be perpendicular to the plane through the given straight lines. 19. If the opposite edges of a tetrahedron be equal two and two, prove that the faces are acute-angled triangles. 20. Prove that a tetrahedron can be formed having each of its faces equal in all respects to a given acuteangled triangle. 21. Within the area of a given triangle is described a triangle, the sides of which are parallel to those of the given one. Prove that the sum of the angles subtended by the sides of the interior triangle at any point not in the plane of the triangles is less than the sum of the angles subtended at the same point by the sides of the exterior triangle. 22. The three angles at the vertex of a tetrahedron are together less than the angles subtended by the opposite edges at any point within the solid. 23. The locus of a straight line which is always | to a given straight line and passes through another given straight line is a plane. 24. The locus of a straight line which is always 1 to a given plane and passes through a given straight line in that plane is a plane. 25. The locus of a straight line which is always 1 to a given plane and passes through any straight line is a plane. 26. The locus of the feet of the perpendiculars let fall from points in a given straight line upon a given plane is a straight line. This locus is called the projection of the given straight line upon the given plane. CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS, |