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circumference, one on each side of the diameter drawn through the point.

If in any diameter two points be taken equidistant from the centre, and four equal lines be drawn from these points to the circumference, the figure formed by joining the extremities of these four lines taken in order will be a rectangle.

3. If a straight line touch a circle and from the point of contact a straight line be drawn meeting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles in the alternate segments of the circle.

Two circles whose centres are A, B, touch externally in C; a common tangent meets the circles in P, Q. Show that the square on PQ is equal to four times the rectangle AC, CB.


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The solution must be given at such length as to be intelligible to

the Examiner, otherwise the answer will be considered of no value. SECTION V. (a.) If a=3, b = 4, c= = 27, find the value of 3ab + ac +47a2b -- 39130 + + 62. (6.) Express in the ordinary notation :

7 x 107 + 7 x 105 + 7 x 103 + 7 x 10. (c.) Simplify the expression :

50 - 6a - {3 (26 — c) + 4 (a -26)-6(2a-c). (d.) Prove the formula am X a” = am+n. (e.) Multiply 23 — m'y - 2y3 by 203 + xy - 2ys.

(These form one question.) Section VI. 1. Resolve into factors 22 + 4x – 21; 12204 + 1læạy2 — 15yo; a* + a2b2 + 64. 2. Simplify the expression: 22 (y — )2 y - (2-2) 22 - (x y)2

+ (a + )-4 (y + x) 22 (y + 2) - 2 3. Prove the following identity; (b -c)3 + (c-a) + (a - b)=3 {a'(c - 6) + b (a - c) + c* (b-a)?

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SECTION VII. Solve the equations

2 x + 1 3 -1 5x + 3 1.


= 8.

2. (a + a) (+ b)

x (a + c) (x + d).
8 + a + b X +c+d



4. 2? + 2x + 5 =

acea + 2.0

(These form one question.) SECTION VIII. 1. Find a number such that when it is divided into 4 and into 3 equal parts, the continued product of the former shall equal eighty-one times the continued product of the latter.

2. Two trucks whose wheels are of different sizes are in motion; one makes 2002 revolutions more than the other in 7 miles, whereas if its wheels had been twice as large in circumference, it would have made 770 less than the other : find the circumference of the wheels.


MENSURATION. SECTION IX. 1. A rectangular park, one side of which is twice as long as the other, contains 500 acres : how much ground will be occupied by a road 15 feet wide, running round it ?

2. Two equal chords, 15 inches in length, are drawn from a point in«the circumference of a circle, and form an angle equal to two-thirds of a right angle : find the area of the circle to three places of decimals.


Females. The solution must in every instance be given at such length as

to be intelligible to the Examiner, otherwise the answer will be considered of no value. SECTION I. Add together nine hundred and seventy thousands seven hundred and nine; three millions thirty thousands and thirty-three; fifty-nine thousands

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five hundred and ninety; eight millions eight bundred and eight thousands and eighty-eight; sixty-seven thousands and seventy-six. Multiply thesum by seventynine, and divide by eighty-four.

Section II. Divide 6061 tons 11 cwt. 2 qrs. 18 lbs. 5 ozs. 5 drams by 69; and prove your sum by multiplication. SECTION III. Work out the following bill of parcels :

137 reading books at 5 d. each.

at ls. 9 d. each.
31 cedar pencils at 9d. a dozen.
132 reams of paper at 31d. a quire.
250 steel pens at 3d. per dozen.

7 dozen slates at 2 d. each. SECTION IV. Find, by Practice, the valne of 191 acres 3 roods 37 poles of land at £42 38. 4d. per acre.

SECTION V. 1. What sum of money, invested in 3 per cent. stock at 873, would produce 50 guineas per annum ? and what increase of income would be obtained by selling out the stock at par and investing in stock at 105 paying 55 per cent. ?

2. A person has an income of £122 108. from 3} per cent. consols; he sells at 92, and invests his principal in stock at 95, which yields 54 per cent. : what will his income be?

SECTION VI. 1. The carriage of one lot of goods weighing 3 tons 1 cwt. 16 lbs. cost £1 158. 8d. : wbat would be the cost of carriage for 8 other lots weighing each 1 ton 13 cwt. 8 lbs. 2 oz. ?

2. If 18 men can dig a trench 17 yards long and 2 yards broad in 17 days, working 5 hours a day, how many men will dig a trench 12 yards long and 4 yards broad in 9 days, working 8 hours a day?

SECTION VII. 1. Arrange the following fractions in order of magnitude :1s, ii, , s, and .

3 2. Multiply 3. by 73, and divide

53 together their sum and difference.

SECTION VIII. 1. Find the value of •732 + 4.96 + 2 X 2:395+ 13.798543 - 21131.



by , and add

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2. Multiply 3:05 by 203, and 1787 by :373, and divide 14:4 by .012.

SECTION IX. 1. Find the expense of carpeting a room 31 ft. 6 in. long, and 16 ft. 6 in. broad with carpet of yd. wide at 58. 3d. a yard.

2. What would be the cost of covering a space 14 yards by 10 with lead of 23 lbs. to the square foot at 4d. a pound?



SCHOOL MANAGEMENT. THREE Hours allowed for this Paper with that on Music. Those who are or have been Pupil Teachers are not to answer

more than one question in any Section. Candidates who have not been Pupil Teachers may answer any seven questions they think fit, except in Section VII., from which only one

subject should be selected for notes of a lesson. No Candidate is to answer more than seven questions.

SECTION 1. 1. For what number of children were desks provided in your school ? Describe the desks used, and state how much space was allowed for each child for writing. What lessons were given out of desks? or in a standing position ? Why were they so given ?

2. What movements of drill were required in your school for entering or leaving the room, and for changing class ? What are the objects of school drill ? Show that it has an influence upon the character of children and their behaviour out of school.

3. In what different shapes may classes be arranged for reading, arithmetic, and geography lessons ?

State which shape you consider best for each purpose, and why you would employ that arrangement. What should be the position of the teacher in regard to his class ?

(For Females only.) 4. Name the parts of a boy's shirt of simple pattern, the proportions required in cutting it out, and the various kinds of work required for each part.

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