Examination Papers of Training Colleges. 441

MUSIC.

Section 1.

Write out the substance of a lesson on "sharps and flats."

Section 2.

66

1. Explain the terms "Ima volta," "24a volta," " da capo al segno," "bis," ❝ gva"

2. Write down the principal words used to express the pace at which music is to be performed, and any derivatives from these words which are in common

use.

3. What is meant by "inversion?" what law is to be observed in all inversions ?

Section 3.

1. Explain, as you would to a class of children, the terms " melody," "harmony," "root," "fundamental base.”

2. What note in a scale is called "the leading note;" and why? Can the leading note have a common chord?

3. Can all the sounds of the common chord move at the discretion of the performer? If you know any rule which regulates one or more of them, state it.

Section 4.

1. If you had a class about to commence the study of harmony, to what points would you direct attention, in the first instance?

2. Make common cadences, with the chords in different positions, in the key of C major, some with, and some without, inserting the

dominant.

3. How is the chord of the diminished seventh produced? example.

upon the

Give an

1. How would you organise a school of 100 children, from 7 to 13 years of age, supposing that you had two pupil-teachers in their second and fourth

years respectively? Draw a plan of the school-room you would prefer, showing the arrangement of the classes, and of the forms and desks. What subjects would you yourself teach, and what would you assign to your pupilteachers?

2. What different methods have been devised for organising elementary schools? Illustrate your descriptions of these by diagrams, state which of them you yourself prefer, and the reasons for that preference.

3. What objects should specially be kept in view in the organisation of a school? What are the advantages resulting from a good organisation, and what are those elements of a good school which no organisation, however good, will secure?

Section 2.

1. Show the divisions of the page of a register, by which the date of the transfer of each boy in a school from class to class may be recorded and easily referred to. What would be the advantages of using such a register? What other means could you devise for recording the progress which each child is making?

2. What expedients should be adopted to secure a regular attendance of the children in a school? What are those qualities of the master which are most likely to promote this regular attendance?

3. Describe some of the characteristic defects of teaching in elementary schools.

Section 3.

1. What are the characteristic dangers of the schoolmaster's profession; 1st, with reference to himself; 2nd, with reference to his scholars ?

2. In what respects may the selfishness of a teacher be prejudicial to the interests of his scholars and to his own? What facilities are afforded him for the indulgence of it?

3. What ground is there for having faith in education; first, from Scripture; secondly, from reason? Considering the education of children to be going on partly at home and partly at school, state in what respects each of these two kinds of education has resources peculiar to itself, and advantages over the other. What reasonable ground is there for confidence in a good school education, even if it be counteracted by the education of the home?

ALGEBRA.

(FOUR HOURS ALLOWED FOR THIS PAPER, WITH THAT ON HIGHER MATHEMATICS.)

1. A. and B. jointly have a fortune of 9,8007. A. invests the sixth part of his property in business, and B. the fifth part, and each has the same sum remaining. How much had each?

At what rate per cent. would the present value of a debt of 450l. payable in 5 years be the same as that of 400l. payable in 3 years.

N. B.-Work this problem, if you can, supposing compound interest: if not, supposing simple interest.

3. Three labourers are employed on a certain work. A. and B. would, together, complete this work in a days; A. and C. would require b days to complete it in, and B. and C., c days. In what time would each of them finish it alone; and how long will they take when all working together?

Section 5.

1. I buy a piece of cloth for 31. If there had been 3 yards less in it it would have cost a shilling more per yard. How many yards did it measure? 2. What two numbers are those whose difference multiplied by the difference of their squares is 160, and their sum, multiplied by the sum of their squares, 580?

3. The joint capital of two partners is 2,000l. One of them withdrew at the end of 12 months, and received for capital and profit 1,040. The capital and profit of the other amounted at the end of 17 months to 1,710. Supposing the same interest to have been made during the whole of this time on the capital invested, and allowing simple interest, how much did each invest?

444

Appendix C.

EUCLID.

(FOUR HOURS ALLOWED FOR THIS PAPER.)

Section 1.

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

2. In any right angled triangle, the square which is described on the side subtending the right angle is equal to the sum of the squares described upon the sides which contain the right angle.

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

Section 2.

1. If in a circle two straight lines cut one another, which do not both pass through the centre, they do not bisect each other.

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

3. To inscribe a circle in a given square.

Section 3.

1. In a right angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another.

2. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about their equal angles reciprocally proportional.

3. Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Section 4.

1. Upon a given base to describe an isosceles triangle equal to a given rectangle.

2. To find a point within a triangle, so that lines drawn to the angles shall divide the triangle into three equal parts.

3. Show that the lines which bisect the angles of a parallelogram form a rectangle.

4. The perpendiculars let fall from the three angles of any triangle on the opposite sides intersect each other in the same point.

MENSURATION.

Section 1.

1. Prove the rule for determining the area of a triangle, having given the base and the perpendicular upon it from the opposite angle.

2. Prove the rule for finding the area of a triangle, having given the sides. 3. Prove the formula for determining the volume of earth taken from na excavation, known as the Prismoidal Formula.

Section 1.

1. What is the area of a room 16 ft. 7 in. long, and 13 ft. 5 in. wide? Prove each step in the operation, and interpret each in the result.

Examination Papers of Training Colleges. 445

What would be

2. There is a goblet of gold the price of which is 100%. the price of a similar goblet which would contain twice as much? The thickness of the gold in the two goblets is to be the same.

3. A circular ring is to be constructed with a given quantity of iron so as to have a given surface; the section of the iron of the ring is to be square; determine its dimensions.

HIGHER MATHEMATICS.

Section 1.

1. Find the 7th term of the series

−1, −1, − 1, &c.

2. What is that arithmetical series having 29 terms, whose first term is 3, and the last 17 ?

3. Given the first term, the last term, and the sum in a geometrical progression; it is required to find an expression for the number of terms.

Section 2.

1. In how many different ways can the letters a, b, c, d, e, f, g, be written after one another? How many of these begin with fg?

2. A farmer proposes to lay out 881. 10s. in purchasing two kinds of sheep, the average price of one kind being 21s. and of the other 31s. per head. În how many different ways can he make up his flock of these two kinds of sheep, so as just to lay out that money?

1+x

3. Expand into a series ascending by powers of x, by the method of (1-x)3 indeterminate coefficients.

Section 3.

1. What will a capital of £a, invested at r per cent. compound interest, amount to in n years, supposing £b to be taken from it annually?

2. A usurer lent 600l. on good security, on condition of being paid back 800l. at the expiration of 3 years. What interest did he take per cent., allowing compound interest?

3. Prove the binomial theorem in the case in which the index is a positive integer; and apply it to determine the middle term of the expansion of

Section 4.

1. Define the logarithm of a number, and show that the logarithm of the quotient of two numbers is equal to the difference of their logarithms. 2. Show that

Cos (A-B) Cos A Cos B+ Sin A Sin B.

=

3. Show that if a, b, c be the sides of a plane triangle, and S half their sum, (S-b) (S-c) and if A be the angle opposite to a, then Tan A = S(S-a)

Section 5.

1. Explain fully what is meant by the differential coefficient of a function, and show how to differentiate the quotient of two functions.

2. Prove Taylor's theorem.

3. Investigate expressions for the area of a parabola, and for the solid content of a spheroid.

1