Section 2.

1. Prove that parallelograms upon equal bases and between the same parallels are equal to one another.

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

3. Prove that if the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle.

Section 3.

1. Prove that 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.

2. Prove that if a straight line be divided into two equal and also two unequal parts, the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section.

3. Prove that in obtuse-angled triangles, if a perpendicular be drawn from either 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.

Section 4.

1. 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.

2. Describe a square that shall be double a given triangle. 3. Prove that the diagonals of a parallelogram bisect each other.

ALGEBRA.

Section 1.

1. What is the approximate value of

4 a √ (b2 ac) + √(4 ac+ e2)
a being = 6, b5, and c = 4.

and subtract (a + b) from 1 (a — b)
—— 3 x2 y2 + 3 y3 by 5 x3 + 3 x2 y2— 5 y3

3. Multiply 4 x

and divide a3

x3 by a + x.

Section 2.

1. Reduce the following fraction to its lowest terms:

2. Add together

36

6 x

+

5 c

9 y

12 ax + 9 x2

2 5 a
+ + + +
2 3 7 4

3. Required the square root of 4 a2

and the fifth power of (a + b) by the binomial theorem.

Section 3.

1. What is the value of x in the following equation?

2. The first term of an arithmetical series is 1, the common difference 2, and the sum of the series 55; required the number of terms, with the rule for finding them.

1. If 6 be added to a certain number, and the sum multiplied by 7, the product will be 63; required the number.

2. The difference of two numbers is 3, and 3 times the less is equal to twice the greater; required the numbers.

3. Explain the different methods of elimination in equations of two unknown quantities; and solve the following:

7

y

+ 7y99 and +7 x = 51

4. A and B commence trade with different sums of money; A gains £40, and then his capital is twice B's; A then loses £80, and B gains £70; and then B's capital is 6 times A's. How much had each at first?

MENSURATION.

Section 1.

1. Find the area of a rhombus whose side is 5 ft. 7 in., and perpendicular height 4 ft.

2. A triangular field, 738 links long and 583 links in the perpendicular, produces an income of £12 a-year; at how much an acre is it let?

3. How many square feet are there in a circle whose circumference is 6.2832 feet?

Section 2.

1. How many gallons will a cistern hold whose depth is 23 ft. 6 in. and diameter 4 ft. 9 in.?

One imperial gallon contains 277.274 cubic inches.

2. A triangular gable, 18 feet high, of 1 brick thick, is raised on an end wall, 20 ft. long and 30 ft. high, of 2 bricks thick ; what is the cost of the whole at £4 a standard rod?

3. If from a right cone, whose slant. height is 30 feet, and circumference of base 10 feet, there be cut off by a plane parallel to to the base a cone of 6 feet in slant height, what is the surface of the frustum remaining?

Section 3.

1. A field in the form of an equilateral triangle contains half an acre; what must be the length of a tether fixed at one of its angles to enable a horse to graze exactly half of it?

2. Suppose the ball on the top of St. Paul's to be 6 ft. in diameter, what would the gilding of it cost at 34d. per square inch? 3. What weight of powder will fill a cylinder whose height is 3 ft., and diameter of base 10 inches?

Section 4.

1. Describe the theodolite, and explain its uses.

2. Explain how the changes of level are ascertained preparatory to constructing a sewer.

9. What is the nature of the different errors to which the instruments used in levelling are liable?

MECHANICS.

Section 1.

1. A cubic foot of water weighs 62.5 lbs. What amount of work will it take to raise 20 cubic feet 12 yards high?

2. What must be the horse-power of an engine to pump 87 cubic feet of water per minute from a depth of 60 fathoms ?

3. Required the horse-power of an engine which moves a train whose weight is 100 tons, the speed 20 miles per hour, and the friction 8 lbs. per ton.

Section 2.

1. How many kinds of levers are there? Give familiar examples, and show the relative position of the power to the weight, and the mechanical advantage in each case.

2. How far from the fulcrum must a power of 7 lbs. be applied so as to balance a weight of 126 lbs. placed at a distance of 2 inches from it?

3. The length of an inclined plane is 80 ft., the height 2 ft., the weight 2 tons; what power must be applied to move the body up the plane, the resistance of friction being 1-30th of the load.

Section 3.

1. Let there be two moveable pullies, weighing 4 lbs. each ; what must be the power when the weight is 180 lbs. ?

2. What power must be applied to the wheel and axle, the radius of the wheel being 20 inches, and that of the axle 4 inches, to support a weight of 400 lbs. ?

3. Explain the theory of falling bodies, and give particular examples to show the relation between the space, the time, and the velocity.

Section 4.

1. Explain the principle of the parallelogram of forces; and state what is meant by the resolution and composition of forces. 2. Explain what is meant by the centre of gravity, and show how to determine the centre of gravity of a triangle, a pyramid, and any four-sided figure.

3. State the condition to be fulfilled in order that any number of forces not acting on a single point may balance each other. What is the nature of the proof of this principle?

POPULAR ASTRONOMY.

Section 1.

1. What is the exact length of the year?

2. What is the Metonic cycle?

3. What modes have in different ages been adopted to remedy the inconveniences that might arise from the year not consisting of an integral number of days?

Section 2.

1. Name the principal constellations in the northern hemisphere, and describe how you would find them in the sky.

2. What is a nebulous star? Mention some of the most remarkable.

3. Do any of the stars vary in brightness? What account has been given of this?

Section 3.

1. If the moon is attracted by the earth, what prevents it from falling like a stone?

2. At what rate does light travel? And how has that rate been determined?

3. To what practical purposes have the eclipses of Jupiter's moons been applied; and what difficulties attend such an application ?

Section 4.

1. Why does the moon always present the same face to us? 2. What is the exact form of the earth's orbit? How would you describe it upon paper?

3. Mention some of the principal comets, the kind of orbits they describe, and the length of time employed in their revolutions.

NOTES OF A LESSON.

Write out the heads of two lessons, supposed to be given to a section of thirty or forty of your eldest children, selecting one subject out of each of two of the following lists.

1. OBJECT LESSONS.-On the human hand or foot; the horse, ox, sheep, or dog; the bee or butterfly; the oak or holly; the