MISCELLANEOUS EXERCISES IN PRACTICAL SOLID GEOMETRY. (A.) 1. Project a line 3 inches long when parallel to the horizontal plane, and at right angles to the vertical plane, its height above the ground being 2 inches, and distance from the vertical plane 2 inches. 2. A and B, 3 inches apart, are the plans of two points, of which A' is 2 inches, and B' 3.5 inches, above the paper. What is the length and inclination of to the paper, of the line A'B' ? 3. Draw the plan and elevation of a black-board 4 feet square, suspended 3 feet above the floor of a schoolroom to which it is parallel and against the wall. [Scale, 2 feet to an inch.] 4. Draw the plan of a line 4 inches long when inclined at 45°, and an elevation of it on any vertical plane not parallel to the line. 5. Project a piece of straight wire 3 feet long, which is fixed in the wall at right angles to it, and 6 feet above the ground to which it is parallel. [Scale, 2 feet to an inch.] 6. The plan of a line is 1.5" long, and its elevation is 3". The projectors of its extremities are 1" apart, measured along wy. What is its true length and inclination ? 7. Project a square prism, one end of which rests on the horizontal plane, and one of its upright faces is parallel to the vertical plane. The height of the long edges is 8', and of the end edges 4'. [Scale, š" to the foot.] 8. Draw the plan and elevation of a point A, which is situated above the horizontal plane, 3" behind the vertical plane, and 3" distant from xy. S 9. Project a triangular prism resting on one of its ends, and having one of its faces parallel to the vertical plane ; its height being 10', and the breadth of each of its triangular edges 4. [Scale, f inch to the foot.] 10. A line AB, 4" long, is inclined 50° to the horizontal plane. Draw its projections when its plan makes an angle of 35° with xy. 11. Project a hexagonal prism resting on the floor of a room having one of its faces inclined at an angle of 45° to the vertical plane, the height of the prism being 8' and the width of each face 4'. [Scale, 6 feet to an inch.) 12. Draw both plan and elevation of a cube of 3" edge, when its base is horizontal, and •5" above the paper ; its horizontal edges making angles of 35° with the vertical plane. (B.) 13. A globe, 10” in diameter, stands on a square table, the edge of which touches the wall of a room. Draw its plan and elevation on a scale of 1' to 1". 14. Draw plan and elevation of a square prism of any size, when its long edges are horizontal, and one of its faces makes an angle of 35° with the paper. 15. Project a cylinder resting on the horizontal plane on one of its ends ; its height is 6', and the diameter of its base 3. [Scale, ' to the foot.) 16. Draw both plan and elevation of a tetrahedron of 2" edge, when its axis is vertical. 17. Project a cone resting on the horizontal plane, its vertical height being 6' and the diameter of its base 3. [Scale, &" to the foot.] 18. A cone, base 1.5" radius, 3” high, is cut by a plane at 70° with the axis ; the centre of the section being 2" above the base. Show the plan of the cut. 19. Project a tetrahedron of any size, having one of its faces resting on the horizontal plane, and one side inclined at an angle of 40°. 20. Draw plan and elevation of a square pyramid, base 1" side, height 4" when one of its long edges is inclined 20° to the paper. 21. Project a pentagonal prism which stands on the floor on one of its ends, having a hidden face parallel to the horizontal plane. Height 5 inches, and width of face 14 inch. 22. A hexagonal prism, base 1" edge, and 3" long, has its axis horizontal, one of its faces being inclined 15° to the paper. Draw both plan and elevation, and a second elevation upon a vertical plane, making an angle of 45° with the plan of the axis. 23. Draw the elevation and plan of a cone, the height of the elevation being 3 inches, and the width of the base 1.5 inch. 24. A pyramid having for its base a square 3" side, and its axis 3.5" long, rests with one face on the horizontal plane. Draw its plan, and a sectional elevation on a vertical plane represented by a line bisecting the plan of the axis, and making an angle of 60° with it. (C.) 25. AB is the elevation of a line 4 feet long, which is parallel to the horizontal plane, but inclined to the vertical plane. Project its plan and determine the angle at which it is inclined. [Scale, }" to the foot.] 26. The horizontal and vertical traces of a certain oblique plane make angles of 40° and 80° respectively with xy. Assume any point above the base line as the elevation of a point contained by this plane, and determine its plan. 27. AB is the plan of a line 4 feet long, which is parallel to the vertical plane, but inclined to the horizontal plane. Project its elevation, and determine the size of the angle of inclination. [Scale " to the foot.] 28. Draw a line parallel to xy, at a distance of 1.5" from it. Consider this as the horizontal trace of certain plane inclined 40° to the horizontal plane, and determine the vertical trace. 29. Project an equilateral triangle, its surface being inclined at an angle of 45° to the vertical plane, with its base parallel to the horizontal plane. 30. Draw two parallel planes, inclined 50° to the horizontal plane, and l" apart; their horizontal traces to make angles of 40° with xy. 31. Project a regular hexagon at an angle of 40° to the vertical plane, its axis being parallel to the horizontal plane. 32. Draw the traces of a plane inclined 75° to the horizontal plane, and 35° to the vertical plane. 33. Project a hexagonal prism, resting on one of its solid angles ; its axis being inclined to the horizontal plane at an angle of 60°, but parallel to the vertical plane. Its height is 8', and the width of each of its faces 4'. [Scale š" to the foot.] 34. A square has its surface inclined 45°, neither of its sides being horizontal. Draw plan and elevation. 35. A square prism, base 2" by 4" long, has one of its rectangular faces inclined 40°, the diagonal of that face being horizontal. Draw plan and elevation. 36. The axis of a square pyramid, base 1:5" side, 4" long, is inclined 60°, one edge of the base being horizontal. Show th true shape of a horizontal section bisecting the axis. QUESTIONS IN PLANE GEOMETRY. Section 1. (1.) Define a point. (2.) What is the true mathematical point ? (3.) Define a line. (4.) What are the ends of lines called ? (5.) What is the point of intersection ? (6.) Name the different kinds of lines. (7.) What is a straight line ? (8.) When is it said to be produced ? (9.) What is a curved line? (10.) Name the directions that a curved line may have. ·(11.) What is a horizontal line ? (12.) What is a vertical line? (13.) What is an oblique line? (14.) How many oblique lines may there be ? (15.) What are parallel lines ? (16.) Name the different kinds. (17.) What is an angle? (18.) On what does its magnitude depend ? (19.) Name the different kinds. (20.) When is a straight line perpendicular to another? (21.) Is a perpendicular line always vertical ? (22.) What is a right angle? (23.) Why is it made the standard for comparing other angles ? (24.) What is an obtuse angle? (25.) What is an acute angle? (26.) What is a circle ? (27.) Distinguish between circle and circumference. (28.) What is an arc? (29.) Define radius. (30.) What is a diameter ? (31.) What is a semicircle ? (32.) What is a tangent? (33.) Into how many parts is the circumference of every circle divided? (34.) What are the parts called ? (35.) How many degrees in a quadrant? (36.) What relationship is there between the angles at the centre of a circle and the arcs on which they stand ? (37.) What is a minute ? (38.) What is a second ? Section 2. (1.) What is Euclid's definition of a figure? (2.) What is Euclid's definition of rectilineal figures? (3.) What is a triangle? (4.) How is it the most simple of all rectilineal figures ? (5.) Why is it sometimes called a trilateral ? (6.) If a rectilineal figure has six sides, how many angles has it? (7.) How many kinds of triangles are there? (8.) From what are they named ? (9.) What is an equilateral triangle ? (10.) What is an isosceles triangle ? (11.) What is |