1. Define a plane angle, a right angle, and a rhomboid. Explain the terms axiom, problem, hypothesis, reductio ad absurdum.

2. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles on the other side of the base shall be equal.

3. If two straight lines cut one another, the angles which they make at the point where they cut, are together equal to four right angles.

4. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.

5. Parallelograms upon equal bases and between the same parallels are equal to one another.

6. If a straight line be divided into any two parts, the squares on the whole line and 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.

7. If ABC be a triangle, and AD, BE, be drawn perpendicular to BC, AC, and intersect in F, the rectangle contained by AF, FD, is equal to that contained by BF, FE.

II. ALGEBRA.

1. Add together x-ty, Ey-x, 16y—1x.

2. (i.) Multiply 3a2-7ax+9y2 by 4a2+7ax-3y'.

(ii.) Divide a +1 by a3 +a√2a+1.

3. Reduce

7x3-11x2 +4

11x3-7x-4

to its lowest terms.

2

X

4. Simplify the fraction [x- (x−})]

:

(3x+1)-(2x-1)

5x-3x+1

5. Solve the equations :

√2x-1+ √x-1 2

6. Gold loses 5 per cent. when weighed in water, silver 9; find the weight of each metal in an ornament weighing 30 oz., which loses 18 oz. when weighed in water.

1. Explain how Force is measured, first, statically, next, dynamically. State the law which expresses the relation of these two measures of force to each other.

2. Enunciate the proposition of the Parallelogram of Pressures.

3. Explain (with the help of a diagram) how the tension of the string of a drawn bow urges the arrow. Show clearly that the pressure on the arrow becomes greater the farther the bow is drawn, even if the tension of the string remains unaltered.

4. Define the term Composition of Motions. Explain (with a diagram) how a steam-boat might cross from the south bank of a river flowing westward, to a point due north on the opposite bank, by keeping on a compass course between north and east.

5. Define a lever, and state the condition for the equilibrium of pressures acting on a lever.

6. A large pair of scales is so placed that its pans are over two holes in the ground, with the surface of which they are level when the beam is horizontal. A loaded wheelbarrow rests with its wheel on one scale and its legs on the ground, and is balanced in that position. A person not strong enough to lift the barrow in the usual way by the handles, attempts to do so; shew that the scale in which the wheel is will descend on his making the attempt.

7. What is meant by the inertia of matter? Illustrate your answer by reference to the intermixture of the components of a liquid when a bottle partly filled with it is shaken vertically.

8. Show, by reference to the laws of falling bodies, that the nomentum acquired by a body in falling from a certain height is equal to the momentum acquired by a body of twice the weight, in falling through one-fourth of the height.

In driving a pile by letting a weight fall on its head, which is the more advantageous way (as regards the driving of the pile) of increasing the momentum, to raise the weight to a greater height, or to increase the weight? Give the reasons for your answer.