2. Draw a ground-plan of your school-room, showing the position of the doors and windows in it, of the teacher's and other desks, of the seats and benches, with the arrangement of the different classes.

Section 4.

Write plain notes of two lessons on one of the following subjects, attending carefully to the circumstances of the case, and stating the length of each lesson.

1. Geography :-
Mountains of England
2. English Grammar :-
Syntax.

3. English History:---
Norman Conquest.

4. Natural History :1. The Whale. 2. Mammalia.

5. Object Lesson :Glass.

(1.) Collective lesson to three lower classes. (2.) To 1st class in a good school.

(1.) Elementary lesson to young children.
(2.) To upper class.

(1.) Collective lesson to three classes.
(2.) Lesson to pupil-teachers in their third

year.

Gallery lesson to young children.
Lesson to 1st class.

((1.) For young children.
(2.) For upper class.

ARITHMETIC.

Section 1.

1. Explain each step in the process of multiplying 6,508 by 4,020. 2. Explain each step in the process of dividing 72,724 by 408, and describe exactly the relation of the remainder to the divisor.

3. Multiply 178. 41d. by 145, and explain the method employed.

Section 2.

1. Find by Practice the value of 251 cwt. 2 qrs. 1 lb. at 17. 17s. ld. per cwt.

2. If 1,200 lbs. can be carried 36 miles for 24s., how many pounds may be carried 24 miles for the same money? Explain each step in the process of working the sum.

3. If 12 men can perform a piece of work in 20 days, what number of men will be required to perform another piece of work four times as great in a fifth part of the time?

Section 3.

1. If the numerator of a fraction be regarded as representing whole numbers, what does the denominator represent?

341

2. Find the value of £4, and find what fraction 2s. 8d. is of 17. 2s. 9d. 3. Multiply 19 by 115, and explain each step of the operation.

Section. 4.

1. How do you multiply or divide a decimal fraction by 10; and why do the methods you adopt answer the purpose?

2. What is the value of 165625 of a ton?

3. Divide 31 by ⚫124689, and give the reason for the correct placing of the decimal point in the quotient.

4. Extract the cube root of 31 to three places of decimals, and explain each operation in the process.

Section 5.

1. What is the difference between "Dr." and "Cr.," and where do you place these terms?

2. What is the "double entry " which gives name to the common plan of book-keeping?

3. What is the nature of the account for " Stock?"

4. When the sale is of various goods in small quantities, how would you keep the accounts, and periodically ascertain the state of the affair?

2. 943 men voted at an election, the successful candidate had a majority of 65: how many voted for each candidate?

3. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6.

3. The square of the greater of two numbers multiplied by the less is 448, and the square of the less multiplied by the greater is 392: required the numbers.

2. Find 7 arithmetic means between 3 and 59, and 3 geometric means between and 32.

3. There are 5 numbers in arithmetical progression; their sum is 25, their product 945. Required the numbers.

Section V.

1. Required the sum and difference of 16 ax and √4ax.

2. Find the square root of 11 + 6 √2.

3. Shew that a ratio of greater inequality is diminished, and a ratio of lesser inequality is increased by adding the same quantity to both its terms.

HIGHER MATHEMATICS.

Section 1.

1. Find the discount on £A payable after n years at r per cent, simple interest.

2. Find the present value of an annuity of £P payable for n years, allowing 7 per cent. compound interest; and adapt the expression to the case in which the annuity is paid by m equal instalments at equal intervals of the year

3. Show that any number and the sum of its digits will leave the same remainder when divided by 9.

4. In how many ways can £1 be paid in half crowns, shillings, and sixpences, the number of coins used for each payment being 18?

Section 2.

1 + x 1. Expand in a series ascending by powers of x, by the method of 2+3x indeterminate coefficients; to four terms.

2. Find the sum of the series

nfinitum.

1

1

1

3. Find the number of permutations of n things taken r together.

ad

4. Prove the binomial theorem in the case in which the index is a positive integer.

Section 3.

1. Explain what is meant when the logarithm of a number to a given base is spoken of, and shew that log, m2 =

n logm.

1

(a − 1)2 +

1

3

(a − 1) 3 . ... and N =

(n - I)

show from the exponential theorem

1

n Na "

and deduce from this equation the value of log, n.

3. Investigate the value of the logarithm of a number in a converging series.

3. In a spherical triangle cos c = cos a cos b + sin a sin b cos C.

4. Cos + √ − 1 sin 4)" = cos n e + √√ − 1 sin n e, n being a posi

tive integer.

Section 5.

1. Find the equation to a circle, a point in the circumference bring the origin, and the axis of the abscissæ being a diameter.

2. Investigate the equation to an ellipse, the centre being the origin an the axis major the axis of the abscissæ.

3. Trace the curve whose equation is y2 = a2

1. Find by the method of increments the differential coefficient of the product PQ, P and Q being functions of the same variable x.

2. Differentiate the following expressions:—

3. Prove Maclaurin's theorem, and apply it to expand sin x in a series ascending by powers of x.

4. Investigate generally the conditions under which a function is a maximum or a minimum.

2. Find the area of the rectangular hyperbola between the asymptotes whose equation is 2 x y = a2, and that of the lemniscata whose pola equation is r2 = a2 cos 2 4.

3. Find the length of the arc of a barabola.

MENSURATION.
Section 1.

1. Prove that the area of a rectangle is equal to the length multiplied by the breadth,

2. Multiply 8 feet 5 inches by 4 feet 7 inches, and explain each step in the working.

3. Prove the rule for finding the area of a trapezoid, and find the area of one whose parellel sides are 4 feet 6 inches, and 8 feet 3 inches, and the perpendicular height 5 feet 8 inches.

Section 2.

1. The sides of a triangular corn field are 150, 200, and 250 yards, required the area, and the expense of reaping the field at 9s. 6d. per acre.

2. The exterior diameter of a metal pipe is 3 inches, the interior diameter 2 inches, what is the area of the circular ring in the section ?

3. The area of a right angled triangle is 24, the three sides are in arithmetical progression, required the sides.

Section 3.

1. Explain the construction and use of the vernier.,

2. A rectangular field having been measured it was found that if it were 5 feet broader and 4 feet longer, it would contain 116 feet more, but if it were 4 feet broader and 5 feet longer it would contain 113 feet more; what are its dimensions?

3. The ceiling and cornice of a square room cost 81., being charged at the rate of 10s. per square yard of the ceiling, and 6s. per linear yard of the cornice; what is the length of the side of the room?

4. Prove the rule for finding the area of a circle.

Section 4.

1. Explain generally the method of finding the difference of level between two distant places by means of a series of sights, and show that the difference of levels between the first and last staves is equal to the difference of the sums of the back and fore sights.

2. Describe the spirit level, and the manner of adjusting and using it. 3. Investigate the prismoidal formula.

GEOMETRY.

Section 1.

1. The angles which one right line makes with another upon one side of it are either two right angles, or are together equal to two right angles.

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

3. If a square described on one of the sides of a triangle be equal to the squares described on the other two sides of it; the angle contained by those two sides is a right angle.

Section 2.

1. If a right 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.

2. If a right line be divided into any two parts, the squares of 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 of the other part.

3. To describe a square that shall be equal to a given rectilineal figure.

Section 3.

1. If a right line drawn through the centre of a circle, bisect a right line in it which does not pass through the centre, it shall cut it at right angles, and if it cut it at right angles it shall bisect it.

2. To describe a circle about a given triangle.

Section 4.

1. The difference between any two sides of a triangle is less than the third side.

2. Divide a square into four equal portions by three straight lines drawn from any point in one of its sides.

3. If two exterior angles of a triangle be bisected, and from the point of intersection of the bisecting lines a line be drawn to the opposite angle of the triangle, it will bisect that angle.

POPULAR ASTRONOMY.

Section 1.

Give one proof, and that the simplest, of one of the following truths of astronomy:

1. That the earth is not a boundless plain as it seems to be; or

2. That the moon does not shine by its own light, but by the reflected light of the sun; or

3. That the earth has an annual motion round the sun.

Describe and explain

Section 2.

1. The apparent motions of the fixed stars to an observer who remains in the same place; or

2. The wandering motions of the planets in the heavens; or

3. The apparent annual revolution of the sun through the heavens. What facts indicate such a revolution?

Section 3.

1. Explain the phases of the moon.

2. Account for the tides. Why does high water occur twice in every 24 hours, and why at different hours on successive days?

3. Account for the eclipses of the sun. Why do they not occur every month? When do they occur? Why is not the same eclipse visible at the same time from all parts of the earth?

4. How is it known that we are at different distances from the sun at