distance of this point (projected at c' in Pr. 63) from the horizontal plane remains unaltered, it must also be in the horizontal line c'd'. Upon these principles all the points indicated by literal references in the present problem are determined ; the curves of penetration resulting therefrom intersecting each other at two points projected

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upon the axial line L'K', of which that marked q' alone is seen. The ends of the horizontal cylinder are represented by ellipses, the construction of which will also be clear on referring to the figure, and they do not require any further consideration.



(A.) 1. Draw a straight line 8 inches long, and divide it into 16 equal parts by continual bisection.

2. Make any line AB, 2 inches long, divide it into four equal parts, and at each end and point of division erect a perpendicular it inch in height.

3. Draw any vertical line AB, 4 inches in length. From the upper extremity, draw a line AC, 3 inches in length, and at right angles to it. Then bisect each of these lines.

4. Construct any triangle, then draw a line perpendicular to the base, and passing through the apex.

5. Construct a scale to represent 20 miles, taking sth of an inch to the mile.

6. Draw a horizontal line AB, 3 inches long. From B, drop a line BC at right angles to AB, 2 inches in length. Then trisect the right angle, and lastly, bisect each of the trisections.

7. Construct an equilateral triangle, and on its three sides respectively construct a square, a hexagon, and a rhombus having an angle of 45°

8. Draw a line to represent 60°, as marked on the side of a map, on a scale of 10° to half an incb.

9. Draw any two parallel lines AB and CD at any distance apart. Find a point E, which shall be equidistant from these lines.

10. Draw a circle of 14 inches radius. Divide it into 6 equal parts, and at each of the points of division, draw a line tangent to the circle.

11. Through any given point A within a circle, whose radius is 14 inch, draw the longest possible chord.

12. There is a stick leaning against a vertical wall, and making an angle of 60° with the ground. Required the angle which the stick makes with the wall.

(B.) 13. Draw a square of 37 inches side, and inscribe in it four equal circles, each touching two others, and two sides of the square.

14. Divide the area of any given circle into six equal sectors, by lines drawn from the centre.

15. Draw a vertical line AB, 3 inches in length. On AB, construct a triangle, having an angle of 50° at A, and an angle of 40° at B. Then state how many degrees the remaining angle contains.

16. Produce a line AB, 3 inches in length, to a point C, so that BC: AB::3:5.

17. Divide a circle into three proportional areas by means of concentric circles, so that the area of the outside circle is three times that of the inside one, and the middle area twice that of the inside one.

18. Construct a triangle having sides respectively of 4 inches, 3 inches, and 2} inches. On the 4 inches side, mark off any four irregular divisions, then divide the 2} inches side proportionately to the divisions on the 4 inches side.

19. Line AB is 3 inches in length, CD is 2 inches, and DE is 1} inch. Find a line, FG, so that CD; AB:: FG: DE.

20. Draw a tangent touching an arc in any given point A, without using the centre.

21. Construct a right-angled triangle, making the hypotenuse twice the length of the base.

22. AB is the mean proportional between two lines 3 inches and 1.5 inch. Find its length.

23. In a given circle whose diameter is 2 inches, inscribe a regular pentagon in two different ways.

24. Draw a vertical line AB, 4 inches in length. Let this line be the altitude of an equilateral triangle. Construct it,

(C.) 25. Draw an equilateral triangle, whose perimeter shall be equal to a square of 1.5 inch.

26. Show the position of a wheat sheaf situated exactly in the middle of a corn-field, which is bounded by six equal hedges.

27. Draw a sector of 3 inches radius, and having an angle of 150°.

28. Construct a right-angled triangle, whose base is 2 inches, the acute angles being in the ratio of 2:1.

29. Prove by illustrations that the angles made by straight lines drawn from the centre of any polygon to the angular points, are together equal to four right angles.

30. Draw a horizontal line AB, 2 inches in length. On AB, as base, construct a triangle, having sides of 2 inches and 3 inches, and find the altitude of the triangle.

31. Determine an equilateral triangle, equal in area to the sum of two squares, having their sides 1 inch and 2 inches respectively.

32. Construct regular polygon whose side AB is the chord of an arc of 45°.

33. Draw the plan of a rectangular field, 400 yards by 250 yards. Mark a point 0, which shall be exactly the centre of the field. [Scale-1 inch to 100 yards.]

34. Draw an equilateral triangle of 2 inches side, and a square equal to it in area.

35. Construct a regular polygon on any given line AB, having the distance from either of its extremities to the centre equal to the side AB.

36. Draw a horizontal line AB, 24 inches in length. Let this line be the altitude of an isosceles triangle. The altitude makes an angle of 15° with one of the sides of the triangle. Construct the triangle.

(D.) 37. Construct a square, an equilateral triangle, and a hexagon. Determine by a squaro the area of the three figures added together.

38. Inscribe in any given circle a triangle that shall cut off equal segments.

39. Draw a rhomboid, letting the shorter side be half the length of the longer side, and one of the angles to contain 60°. Find the centre of the rhomboid, and from it draw a line perpendicular to one of the longer sides.

40. Draw three circles of 1, 13, and 2 inches radii, so that each circle touches the other two.

41. In a given circle 21 inches in diameter, inscribe seven equal circles, six of which shall touch the given circle and a central one.

42. Draw a circle of 1 inch radius. Outside of this circle, find the positions of two points, A and B, which are to be respectively 14 inch and # inch from the circle. A and B are also to be 3 inches from each other. Draw a couple of tangents to the circle, from each of these points.

43. Draw two lines at an angle of 40°. Then draw two circles, each touching the lines and one another, the radius of the smaller one to be 2 inches.

44, In any given square sufficiently large, inscribe a triangle having its two sides equal and its base one inch long.

45. Describe an arc of a circle, and show how its centre may be ascertained if it were not already marked.

46. In any given square inscribe a regular polygon that shall cut off four equal corners of the square.

47. Describe a circle of 13 inch radius. Mark any point A, on the circumference. Inscribe another circle of 1 inch radius within the first circle, and which shall just touch the large circle at point A, tangentially. Then describe another circle of 14 inch diameter, which shall be outside of the large circle, and also just touch point A, tangentially

48. Draw a tangent touching the curve of an ellipse at any given point A, and also a line perpendicular to the curve, from that point.

(E.) 49. Draw the plan of a triangular piece of wood, having sides respectively 3 feet, 2 feet, and if faot. It is required to cut the largest possible circle out of this piece of wood. Show how large the circle would be. [Scale-1 inch to the foot.]

50. Draw a rectangle having sides of 3 inches and 24 inches, and in it inscribe an ellipse,

51. Draw a line AB, an inch long. From B, draw line BC, 2 inches long, and making an angle of 30° with AB. These lines are adjacent chords of a circle ; describe the circle.

52. Within an octagon whose base is 14 inch, inscribe a similar concentric octagon whose base is finch.

53. There is a thin piece of metal in the shape of an isosceles triangle. Its base is 3 inches long, and the angle at the apex is a right angle. Show how to cut this into four smaller equal triangles, all similar in shape to the large triangle.

54. About a given heptagon whose base is 1.5 inch, describe a similar heptagon whose base is 2 inches.

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