5. Describe a square equal in area to the difference of two given squares.

6. The straight line which joins a point without a circle to the centre of the circle bisects the angle between the tangents drawn from that point to the circle.

7. Divide a straight line so that the rectangle contained by the whole line and one of its parts may be equal in area to the square on the other part.

8. If from any point without a circle two straight lines be drawn, one of which cuts the circle and the other touches it ; the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square of the line which touches the circle.

9. Of all chords that can be drawn through a given point within a circle, that is the shortest which is at right angles to the diameter passing through the point.

10. In a given circle to inscribe a triangle equiangular to a given triangle.

11. Any equilateral figure which is inscribed in a circle is also equiangular.

MATRICULATION, JUNE 1875.

Afternoon, 3 to 6.

GEOMETRY.

1. In an isosceles triangle, show that the right line which bisects the vertical angle, bisects the base, and is perpendicular to the base.

2. Given a finite right line of any length: bisect it, and at the point of bisection erect a perpendicular to it.

3. One side of a triangle being supposed longer than another, show that the angle opposite to it is greater than that opposite to the other.

4. In a parallelogram, prove the equality of each pair of opposite angles, and also of each pair of opposite sides.

5. Two or more parallelograms being supposed to have a common base, and to lie between the same parallels, show that they are equal in area.

6. On a given finite right line of any length, as base, construct a rectangle which shall be equal in area to a given

square.

7. A finite right line of any length being supposed cut equally and unequally; show that the rectangle under the unequal parts, with the square on the interval between the points of section, is equal to the square on half the line.

8. In a circle of any radius, show that every diameter subtends a right angle at every point on the circumference.

9. Of all chords of a circle passing through a common point inside the circle, show that the longest passes through, and that the shortest is perpendicular to that passing through, the centre.

10. From a given point outside a given circle, show that two tangents of equal length, and equally inclined to the right line connecting the point and centre, can be drawn to the circle.

II. A tangent and chord being supposed drawn at any point of a circle, show that either angle between them is equal to that in the alternate segment of the circle as divided by the chord.

12. In a given circle inscribe a triangle equiangular to a given triangle, and having an assigned vertex at a given point on the circle.

MATRICULATION, JANUARY 1876.

Afternoon 3 to 6.

GEOMETRY.

1. In a triangle of any magnitude, show that if two sides be equal the opposite angles are equal, and conversely.

2. Two sides of a triangle being supposed given in magnitude, show that the greater the angle between them the greater the third side, and conversely.

3. Through a given point, on either side of a given indefinite right line, draw the perpendicular and the parallel to the line.

4. Two or more parallelograms being supposed to have equal bases, and to lie between the same parallels, show that they are equal in area.

5. A finite right line of any length being supposed cut equally and unequally, show that the sum of the squares of the unequal parts exceeds the sum of the squares of the equal parts by twice the square of the interval between the points of section.

6. Divide a given finite right line of any length, so that the rectangle under the whole line and one part shall be equal to the square of the other part.

7. In a circle of any radius, show that two or more equal chords are equidistant from the centre, and conversely.

8. Through a given point inside a given circle, draw the chord of the circle which shall be bisected at the point.

9. If a quadrilateral be inscribed in a circle, show that the sum of one pair of its opposite angles is equal to the sum of the other pair.

10. If a quadrilateral be circumscribed about a circle, show that the sum of one pair of its opposite sides is equal to the sum of the other pair.

1. In a given circle inscribe a triangle equiangular to a

given triangle, and having its vertex at a given point on the circle.

12. About a given circle circumscribe a triangle equiangular to a given triangle, and having its base on a given tangent to the circle.

MATRICULATION, JUNE 1876.

1. Prove that all right-angled triangles are equal in all respects if the hypothenuse and a side in the one are respectively equal to the hypothenuse and a side in the other.

2. Show how to construct a triangle, having given two angles and the side opposite one of them.

3. Construct an equilateral triangle, having given its altitude.

4. Divide a right angle into six equal parts.

5. Prove that every quadrilateral whose diagonals bisect each other at right angles is a rhombus.

6. A quadrilateral being formed by joining the middle points of adjacent sides of a rectangle, investigate what kind of quadrilateral it is.

7. On a given base construct a rectangle equal in area to a given parallelogram.

8. Prove that a quadrilateral which has two opposite sides parallel is equal in area to a rectangle between the same parallels on a base equal to half the sum of the parallel sides of the given quadrilateral.

9. Define the circumference of a circle; and prove that no straight line can cut it in more than two points.

10. Prove that two angles at the circumference of a circle are equal if they stand upon equal arcs.

II. Prove that the line joining the centres of two intersecting circles bisects the common chord at right angles. 12. In a given circle inscribe a regular decagon.