14. A farmer buys a number of oxen for 200 guineas, and, after losing 4 of them, sells the remainder for £7 a head more than they cost him, and gains by the transaction 20 guineas. What number of oxen did he purchase?

15. Two vessels A and B contain each a mixture of water and wine, A in the ratio of 3: 4, B in that of 5: 6. What quantity must be taken from each to form a mixture which shall consist of 7 gallons of water and 11 of wine?

ALGEBRA.-(B.)

1. DISTINGUISH between the subtraction of algebraical and that of arithmetical quantities.

Subtract 3a2-4ac+2b2-5c2 from 3b2-4ab+ a2 - 2c2.

2. Find the value of 16a2+5bc-6c2+7ad, when a, b, c, d are respectively equal to 3, 2, 5, 1.

4. Multiply b2+2ab - c2+a2 by c2+a2-2ab+b2, and shew that the result may be expressed under the form (aa— b2)2 + c2 (4ab — c2).

5. State the preliminary steps to be adopted in the division of one algebraical quantity by another.

8. A has th share of a ship and sells th of th of his share for £100.

What is the value of th of th of the ship?

9. A is 30 years younger than B, and 5 years back B was twice the age of A. What is A's age?

10. A takes 9 hours to dig a certain trench and B takes 7 hours. How long will A and B together take to dig it?

11. Prove that a ratio of greater inequality is increased and of less inequality diminished by subtracting the same quantity from both its terms.

12. a, b, c, d are in proportion; prove that

14. A horse-dealer buys a number of horses for £180, and after losing 5 sells the remainder for £9 a head more than he gave for them, losing by the transaction £45. What number of horses did he buy?

15. Two vessels A and B contain different mixtures of wine and water, the one in the proportion of 2: 5, and the other in that of 3: 11. What quantity must be taken from each to form a mixture which shall contain 5 quarts of wine and 13 of water?

EUCLID.-(A.)

1. DEFINE a point, a plane rectilineal angle, and a circle.

2. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

3. Draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.

4. Any two sides of a triangle are together greater than the third side. 5. If two triangles have two angles of one equal to two angles of the other, each to each; and one side equal to one side, viz. either the sides adjacent to the equal angles, or the sides opposite to equal angles in each; then shall the other sides be equal, each to each: and also the third angle of the one to the third angle of the other.

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

7. If a parallelogram and a triangle be upon the same base, and between the same parallels; the parallelogram shall be double of the triangle.

8. 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 of the whole line.

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

10. If two circles cut one another, they shall not have the same centre. 11. Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre, are equal to one another.

12. The angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same part of the circumference.

13. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

EUCLID (B).

1. DEFINE a line, a superficies, and the centre of a circle.

2. Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.

3. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

4. If from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

5. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

6. Parallelograms upon the same base, and between the same parallels, are equal to one another.

7. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

8. 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 of the aforesaid part.

9. Divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

10.

If two circles touch one another internally, they shall not have the same centre.

11. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote: and the greater is nearer to the centre than the less.

12. The opposite angles of any quadrilateral figure described in a circle, are together equal to two right angles.

13. If a straight line touches a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.

MECHANICS.

(Trigonometrical methods of solution are not excluded.)

1. DEFINE Force, Density, Mass.

If the density of one substance be 6.7, and of another 7.2; in what proportions must they be mixed to produce a substance of density 6.9?

2. Prove the Proposition of the Parallelogram of Forces, so far as the magnitude of the resultant is concerned.

Three forces represented by those diagonals of three adjacent faces of a cube which meet, act on a point; the resultant equals twice the diagonal of the cube.

3. If two weights acting perpendicularly to the arms of a straight lever, on opposite sides of the fulcrum, will balance, they are inversely as their distances from the fulcrum.

A window-frame is supported in the usual manner by weights, equal respectively to half the weight of the frame; if one string break, find the pressure sustained by a stick, placed to prop up the window in the same vertical line as the broken string, and state the effect of placing it in any other position.

4. If two weights balance each other on a straight lever, when it is horizontal, they will balance each other in every other position of the lever. Why does this proposition not apply to the case of the Common Balance?

5. In a system in which the same string passes round any number of pulleys and the parts of it between the pulleys are parallel, there is equilibrium, when P: W:: 1:n, where n is the number of strings at the lower block.

Shew the advantages of the system of pulleys, where the radii of those in the lower block are as 1 : 3:5, &c. and in the upper 2: 4: 6, &c.

6. Shew how to graduate the Common Steel-yard.

7. The weight (W) being on an inclined plane, and the force (P) acting parallel to the plane, there is equilibrium when P: W:: the height of the plane to its length.

Compare two weights, which support each other on two planes by means of a string passing over a pulley at the common vertex of the two planes, the string being in each case parallel to the plane.

8. If P and W balance each other on the single moveable pulley, where the strings are parallel, and it be set in motion, P: W :: W's velocity in the direction of gravity : P's velocity.

State the general principle of which this is a particular case.

9. Define "Centre of Gravity." Find that of any number of particles in one plane.

Weights of 1, 2, 3 lbs. respectively are suspended from a weightless bar at distances of 4, 6, 7 inches respectively from one end of it, find the position of the centre of gravity of the weights.

10. Find the centre of gravity of a triangle; and shew that it coincides with that of three equal weights placed at the three angular points.

Four triangles are formed by taking any side of a hexagon as a base and joining its extremities with the other angular points; find their centre of gravity.

11. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line drawn from its centre of gravity, falls within or without its base.

A square stands on a horizontal plane; if equal portions be removed from two opposite corners by lines parallel to a diagonal, find the least portion that can be left, so as not to topple over.

EUCLID. BOOKS I. II. III.—(A).

For all Candidates for a B.A. Degree.

1. UPON the same base and on the same side of it, there cannot be two triangles, which have their sides terminated in one extremity of the base equal to one another and likewise those terminated in the other extremity.

2. If a straight line, falling upon two other straight lines, makes the exterior angle equal to the interior and opposite upon the same side of the line; or makes the interior angle upon the same side equal to two right angles; the two straight lines shall be parallel to each other.

If two lines, which meet, be parallel to two others which also meet, the angles included by the two pairs of lines will be equal.