2. Equal triangles on the same base and on the same side of it are between the same parallels.

If a quadrilateral is bisected by each of its two diagonals, it is a parallelogram.

3. In any right-angled triangle the square described on the side subtending the right angle is equal to the sum of the squares described on the sides containing the right angle.

If in Euclid's figure GH, FD, and KE be drawn, the triangles GAH, FBD, KCE will be equal to the given triangle.

4. ABC is a triangle, AD is drawn perpendicular to BC, and AB, AC are bisected in E, F; show that the angle EDF is equal to the angle A, and that the figure AEDF is half the triangle ABC.

SECTION III.

1. If a straight line be bisected, and produced to any point, the rectangle contained by the whole line thus produced, and the part produced, together with the square on half the line bisected, is equal to the square of the line made up of the half and part produced. Describe a rectangle equal to the difference of two given squares.

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

ABC is an isosceles triangle of which AB, AC are equal, BC is produced to D; then the difference of the squares on AD. and AC is equal to the rectangle DC. DB.

3. In a right-angled triangle the square on the perpendicular from the right angle on the hypothenuse is equal to the rectangle contained by the segments of the hypothenuse. (Euclid Bk. iii. not to be used.)

4. The sides of a triangle are 13, 14, 15 inches; find the nature of the largest angle, and the segments made by the perpendicular from it on the opposide side.

ALGEBRA AND MENSURATION.

Acting Teachers are not obliged to take this Paper.

ALGEBRA.

SECTION I.

1. Divide the product of a3+b3 and a-b' by a'-ab+b3. 2. Divide (a−b-c)x2 - (ab+ac-bc)x-abc by x+c 3. Shew that (a + b + c)2 − (a + b−o)2 + (a−b + c)3 -(a-b-c) 8ac.

SECTION II.

1. State clearly the rule for finding the G. C. M. of two algebraical expressions.

Ex. Find the G. C. M. of

2a3+9a2b+7ab2 - 363 and 3a3+5a2b-15ab2+4b3.

SECTION III.

1. Extract the square root of

a2 - 4a2b* + 10ab 12ab+963.

2. Given √6 2.4494897, find the value of

=

3√2-√3

23-2

SECTION IV.

1. Solve the equation ✈ (x − ) + } (x − 3)=‡8.

√a+ x

2. Solve the equation Va++ √a-x

√a+x √α-x

=

b

3. A man bequeaths of his property to his son, of the remainder to each of his two daughters,

of his

property to his widow, and the remainder, £400, to charities. What was his property worth?

2. When is a series said to be in arithmetical progression? Show how to sum the series

a+b, a+3b, a+5b,...........to n terms.

3. The sum of the 1st and 3rd terms of a series in geometrical progression is 20, and the sum of the 2nd and 4th terms is 60; find the series.

MENSURATION.

SECTION I.

1. The sides of a triangle are 39, 42, and 45 feet; find its area in yards.

2. One diagonal of a trapezoid is 20 chains 35 links, and the perpendiculars on it from the opposite angles are 8 chains 24 links, and 7 chains 36 links; find the area in acres, roods, and perches.

3. The legs of a right-angled triangle are 36 and 48 feet; find the length of the perpendicular from the right angle upon the hypothenuse.

SECTION II.

1. How many pieces of paper, 14 yards long and 18 inches wide, will paper a room 24 feet long, 18 feet wide, and 12 feet high?

2. Find the length of the longest straight line that can be drawn in the room in the last question.

3. The side of a square is 10; show that the area of the regular octagon formed by cutting off equal portions from the four corners of the square will be 200 (2-1).

MALE CANDIDATES-SECOND YEAR.

ARITHMETIC, ALGEBRA, & MENSURATION*.

Two hours and a HALF allowed for this Paper.

Candidates are not permitted to answer more than eight of these questions.

1. Find the square root of 39.175081, and the cube root of 91.125.

2. Establish and state a rule for finding the percentage of gain or loss on the sale of any article. Ex. If eggs be bought at 16 for a shilling and sold at 10 for ninepence; what is the gain or loss per cent.?

3. Find the exact sum of 75-804 and 21-36, and also their difference.

4. A sum of £763.5 being put out at simple interest for 6 years amounts to £962-01; what is the rate per cent.?

5. Solve the equations

-z = 11.

3x+4z = 57.

6. If 3 persons can do a piece of work in a, b, and c days respectively; in what time can they do it when they all work together?

7. If the number of combinations of a things taken 4 together be 7 times the number taken 2 together, find the value of x.

8. Given the sum of two numbers equal to 5, and the sum of their cubes equal to 35; determine the numbers.

9. If the square of any odd number be diminished by 1, the remainder is divisible by 8.

10. The nth term of an arithmetical series being (3n-1), prove that the sum of n terms = (3n+1).

*See infrà, p. 58.

11. If the arithmetic mean between two numbers exceed the geometric by 13, and the geometric exceed the harmonic mean by 12; what are the numbers? 12. Shew that the sum of the series

1. 2. 3 + 2. 3. 4 + 3. 4. 5+ &. to n terms = n (n + 1) ( n + 2) (n + 3)

4

13. One side of a right-angled field is 84 chains long, and the difference between the hypothenuse and the other side is 42 chains; find the hypothenuse and the other side, and also the area of the field in a. r. p.

14. A room is 36 feet long, 25 feet broad, and 18 feet high; how many pieces of paper will be required to cover it, each piece being 12 feet long and 18 inches wide?

15. The slant height of a right cone being 20, and the radius of its base 12 feet, find its solid content.

16. The radius of a sphere being 10 feet, find the convex area of its circumscribing cylinder,

GRAMMAR AND COMPOSITION.

SHAKESPEARE'S Henry the Eighth, or BACON's Advancement of Learning (Part of Book the Second).

COMPOSITION.

Every Candidate must perform the exercise in Composition. It is recommended that not more than half an hour be devoted to this exercise.

Write in plain prose :

1. A narrative of the action in Shakespeare's play of King Henry VIII.

Or 2. A short essay on the importance of the study of language.

Or 3. On the characteristics of poetry as contrasted with prose.

Or 4. Mention some of the sciences now cultivated, which do not come into Bacon's review, not having been known or systematized at the beginning of the 17th century; and give a short account of the commencement, progress, and most distinguished students, of one of them.