37. The lines drawn from two vertices to the two opposite sides, and the sum of these sides being given, to construct the triangle.

38. Two sides, and the line drawn from the vertex of the included angle to the middle of the opposite sides, being given, to construct the triangle.

39. To construct a right-angular triangle, the difference between the hypothenuse and one of the sides of which, is equal to the difference between that and the remaining sides, and the area is equal to a given square.

40. To construct a right-angular triangle, one of the sides of which is a mean proportional between the hypothenuse and the other side, and the area is equal to a given square.

41. To place a given triangle upon another so that the vertices of the latter shall fall in the sides of the former.

42. In a given triangle to construct a rectangle, whose area is equal to a given square.

43. In a given triangle, to construct a rectangle, the sum of the sides of which is equal to a given line.

44. In a given triangle to construct a rectangle, the difference between two adjacent sides of which is equal to a given line.

45. In a given triangle to construct a rectangle, whose diagonal is equal to a given line

46. In a parallelogram to construct a square, whose vertices are in the sides of the parallelogram.

47. In a given quadrilateral to construct a parallelogram, whose sides are parallel to two given straight lines.

48. The difference between one of the diagonals and one of the sides of the square being given, to construct the square.

49. The three angles and the sum of the three sides of a triangle being given, to construct the triangle.

50. To describe a circle, which shall pass through two given points, and touch a given straight line.

51. To describe a circle, which shall pass through a given point, and touch two given straight lines.

52. To describe a circle, which shall pass through two given points, and touch a given circle.

53. To describe a circle, which shall pass through a given point, and touch two given circles.

54. To describe a circle, which shall touch three given circles.

55. To describe a circle, which shall pass through a given point, touch a given straight line, and also a given circle.

56. To describe a circle, which shall touch two given straight lines and also a given circle.

57. To describe a circle, which shall touch a given straight line and two given circles.

58. To construct a rectangle, the ratio of two adjacent sides of which is given, and whose area shall diminish by the area of a given square, if the basis and height diminish by given straight lines.

59. To construct a rectangle, the ratio of two adjacent sides of which is given, and whose area increases by that of a given square, if the basis and height increase by given straight lines.

60. To find in lines, the ratio of two given parallelograms.

61. Several similar figures being given, to construct a figure which is similar to each of them and equal to their sum.

62. From a given point in one of the sides of a triangle, to divide it in a given ratio.

63. From a given point within a triangle, to divide it in a given ratio.

64. To divide a triangle into a given number of equal parts, in such a way that the points of division shall be parallel to a straight line, given in position.

APPENDIX.

Containing Exercises for the Slate.

1. The side of a square being 12 feet, what is its area?

2. What if the side is 12 rods, miles, &c.? 3. What is the side of a square, whose area is one square foot?

4. What that of a square, whose area is one square yard, rod, mile, &c.?

5. What that of a square of 4, 9, 16, 25, 36, 49, 64, 81, 100 feet?

6. What is the area of a rectangle, whose base is 50 feet 3 inches, and whose height 10 feet 4 inches 2 seconds?

7. What that of a rectangle whose base is 40 feet 3 inches 9 seconds, and whose height is 12 feet?

8. If the area of a rectangle is 240 square feet 19 square inches, and its basis measures 30 feet, what is its height?

9. What is the height of a rectangle, whose base is 10 feet and whose area is 40 square feet?

10. What is the height of a rectangle, whose basis is 4 feet, and whose height is 3 inches 5 seconds?

11. What is the area of a parallelogram of 10 feet basis and 3 feet 4 inches high?

12. The height of a parallelogram is 5 feet, and the area 40 square feet, what is its basis?

13. The sum of the two parallel sides of a trapezoid is 12 feet and their distance 4 inches, what is the area of the trapezoid?

14. The area of a trapezoid is 24 square feet, and its height is 4 inches 3 seconds, what is its basis?

15. What is the difference between a triangle, whose basis is 10 feet 3 inches and height 9 feet, and a triangle of 3 feet basis and 11 inches height?

16. What is the difference between a trapezoid, the sum of the two parallel sides of which is 14 feet 3 inches and height 9 inches, and a square upon 9 inches?

17. What is the sum of the areas of a triangle of 3 feet basis and 9 inches height, a square upon 14 feet 3 inches, and a rectangle whose basis is 3 feet 2 inches 9 seconds, and height 4 inches 5 seconds?

18. What is the area of a circle, whose radius is 9 inches.

19. What that of a circle whose radius is 10 feet?

20. What that of a circle, whose radius is 9 feet 6 inches 2 seconds?