34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects the parallelogram, i.e. divides it into two equal parts.

35. Parallelograms on the same base and between the same parallels are equal to one another.

36. Parallelograms on equal bases and between the same parallels are equal to one another.

37. Triangles on the same base and between the same parallels are equal to one another.

38. Triangles on equal bases and between the same parallels are equal to one another.

39. Equal triangles on the same base and on the same side of it are between the same parallels.

40. Equal triangles on equal bases in the same straight line and on the same side of it are between the same parallels.

41. If a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram shall be double of the triangle.

42. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to

one another.

44. To a given straight line to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

45. To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.

46. To describe a square on a given straight line.

47. In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides containing the right angle.

48. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides, the angle contained by these two sides is a right angle.

BOOK II.

1. If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line and the several parts of the divided line.

2. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square on the whole line.

3. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts is equal to the rectangle contained by the two parts, together with the square on the aforesaid part.

4. If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.

5. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

6. If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part produced, together with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

7. If a straight line be divided into any two parts, the squares on the whole line and on one of the parts are equal to twice the rectangle contained by the whole and that part, together with the square on the other part.

8. If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, is equal to the square on the straight line which is made up of the whole and that part.

9. If a straight line be divided into two equal and into two unequal parts, the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section.

10. If a straight line be bisected and produced to any point the squares on the whole line thus produced and the part produced are together double of the square on half the line and of the square on the line made up of the half and the part produced.

E

11. To divide a given straight line into two parts, so that the rectangle contained by the whole and one part shall be equal to the square on the other.

12. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by the side on which, when produced, the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle.

13. In every triangle the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle.

14. To describe a square that shall be equal to a given rectilineal figure.

EXAMINATION PAPERS IN EUCLID.

I.

College of Preceptors, Midsummer 1890.

Third Class.

1. What meaning do you give to the terms base, radius, parallelogram?

Write out one Postulate and two Axioms.

2. Euclid I. 2.

Suppose A were on the circumference of the smaller circle used in the construction, where would the vertex of the equilateral triangle fall?

3. Euclid I. 5.

Show that the straight line which bisects the vertical angle of an isosceles triangle also bisects the base. 4. Euclid I. 12.

5. Any two angles of an isosceles triangle are together less than two right angles.

6. Euclid I. 19.

7. Either, Euclid I. 25.

Or, If two straight lines cut one another, and if two of the adjacent angles be bisected, the bisecting lines shall be at right angles to one another.

II.

College of Preceptors, Christmas 1890.

Third Class.

1. Define a straight line, a plane rectilineal angle, and

an equilateral triangle.