7. The sum of two numbers is 16, and the sum of their reciprocals is. What are the numbers?

8. Compute the value of the continued fraction,

of Indeterminate Coefficients, or by the Binomial Theorem.

10. Insert three geometrical means between and 128.

GEOMETRY.

1876.

(EUCLID.)

1. To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.

2. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

3. If a straight line be divided into any two parts, the square 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.

(LOOMIS AND LEGENDRE.)

1. If a straight line, meeting two other straight lines, makes the interior angles on the same side together equal to two right angles, the two lines are parallel.

2. The radius (or diameter) which is perpendicular to a chord, bisects that chord, and also the arc subtended by it.

3. In every parallelogram, the sum of the squares of the four sides is equal to the sum of the squares of the diagonals.

(CHAUVENET.)

1. If two sides of a triangle are respectively equal to two sides of another, but the included angle in the first triangle is greater than the included angle in the second, the third side of the first triangle is greater than the third side of the second.

2. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.

3. Through any three points, not in the same straight line, a circumference can be made to pass, and but one.

1877.

(EUCLID.)

1. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are also themselves equal and parallel.

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

(LEGENDRE.)

1. In an isosceles triangle the angles opposite the equal sides are equal.

2. In equal circles, equal chords are equally distant from the centres; and of two unequal chords the less is at the greater distance from the centre.

3. The area of a trapezoid is equal to the product of its altitude and half the sum of its parallel sides.

(LOOMIS.)

1. If two triangles have the three sides of the one equal to the three sides of the other, each to each, the three angles will also be equal, each to each, and the triangles themselves will be equal.

2. Two parallels intercept equal arcs on a circumference. 3. In any triangle, if a straight line is drawn from the vertex to the middle of the base, the sum of the squares of the other two sides is equivalent to twice the square of the bisecting line, together with twice the square of half the base.

1878.

(EUCLID.)

1. If a straight line falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another.

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

3. 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.

(LEGENDRE.)

1. If two sides of a quadrilateral are equal and parallel, the figure is a parallelogram.

2. (a) To erect a perpendicular to a given straight line, at a given point of that line.

(b) At a point on a given straight line, to construct an angle equal to a given angle.

3. In any triangle, the square of a side opposite an acute angle is equal to the sum of the squares of the base and the other side, diminished by twice the rectangle of the base and the distance from the vertex of the acute angle to the foot of the perpendicular drawn from the vertex of the opposite angle to the base, or to the base produced.

(LOOMIS.)

1. If two triangles have two sides of the one equal to two sides of the other, each to cach, but the included angles unequal, the base of that which has the greater angle will be greater than the base of the other.

2. Through any three points not in the same straight line, one circumference may be made to pass, and but one.

3. The rectangle contained by the sum and difference of two lines is equivalent to the difference of the squares of those lines.

1879.

[Candidates for examination in Euclid may take questions 2, 3, and 5. Candidates for examination in Loomis may take questions 1, 4, and 5. Candidates for examination in Legendre may take questions 2(b), 3, and 6. Candidates for examination in other Geometries may demonstrate as many of the theorems as they can, and do the problem (3) by the methods to which they are accustomed.]

1. The opposite sides and angles of a parallelogram are equal to each other.

2. If a straight line, meeting two other straight lines,

(a) Make an exterior angle equal to an interior and opposite (or remote) angle on the same side; or

(b) Make the interior angles on the same side together equal to two right angles, the two lines are parallel.

3. To draw a perpendicular to a given straight line, from a given point without that line.

4. Parallelograms which have equal bases and equal altitudes are equivalent.

5. If a straight line be divided into any two parts, the square of the whole line is equal (or equivalent) to the squares of the two parts, together with twice the rectangle contained by the parts.

6. The rectangle contained by the sum and difference of two lines is equivalent to the difference of the squares of those lines.

1880.

[Candidates who offer Euclid may take 1 and 3. Candidates who offer Loomis's Geometry or Davies's Legendre may take 1 and 4. Candidates who offer Chauvenet's Geometry may take 2 and 5. Other candidates may prove theorem 1 or 2, and do one of the problems (3, 4, and 5) by the methods to which they are accustomed.]