6. Equal triangles on equal bases in the same straight line and on the same side of it are between the same parallels.

Show that the figure formed by joining the middle points of the sides of a quadrilateral is half of the quadrilateral. 7. The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.

If one of the parallelograms about the diameter be double of either of the complements, the other parallelogram will be half of either of them.

8. If the 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.

Show that in an equilateral triangle, the square on the base is less than the sum of the squares on the sides.

9. 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 on the aforesaid part.

Show that this proposition in a particular case of proposition I. of the Second Book.

10. To divide a given straight line into two parts so that the rectangle contained by the whole and one of the parts may be equal to the square of the other part.

Show that the parts obtained by solving the question algebraically are incommensurable with the whole line and with each other.

11. Show that the straight lines drawn from the angles of a triangle to the points of bisection of the opposite sides meet at the same point.

12. Show how to trisect a given finite straight line.

BRITISH HISTORY.

Candidates are not to answer more than EIGHT questions.

1. Give a brief account of three of the following persons :Caradog (Caractacus), Boadicea, Agricola, Augustine, Dunstan, Edric the Traitor, Godwin, Simon de Montfort, Wallace and Owain Glyndwr.

2. Of what historical events do the following dates remind you:-84, 878, 1016, 1314, 1415, 1588, 1605, 1660, 1688, 1745, 1798, 1815, 1837?

3. Which of the kings of England were termed the Great, the Martyr, the Ironside, the Unready, the Harefoot, the Confessor, the Rufus, the Lionhearted, the Lack-land, the English Justinian? Give the dates of their rule and explain the circumstances which gave rise to their appellations.

4. Wales has exercised a powerful influence over the affairs of England; give particulars of this influence.

5. Give the names and dates of the kings of Scotland who invaded England, and of those who were imprisoned in England.

6. Relate the circumstances that led to King John's signature to Magna Charta. What sections of society did the Charter affect? Write the two most important of its articles.

7. Describe the leading features of the French campaign of Edward III.

8. Give some account of the condition of the peasantry of England from 1350 to 1550. In whose reign did our modern poor laws originate?

9. What led to the Wars of the Roses? How were these wars brought to a close ?

10. What happened in Ireland during the reigns of Elizabeth, James I., and William III.?

11. Give an account of the Regents of Scotland.

12. How do you account for the arbitrary character of the Government of Charles I.?

13. Show the descent of William I., Henry VII., James I., William III., George I.

14. Describe fully one of Marlborough's campaigns.

15. The reign of George III. has been described as "the most eventful period in the annals of mankind." Can you justify this description?

16. Mention the names of the leading statesmen of the reigns of George II. and William IV. Describe the career of the one who has excited your sympathy.

17. Mention the causes which led to the Indian mutiny, and the chief incidents of that outbreak.

GEOGRAPHY.

Candidates must not answer more than EIGHT questions. Question 1. (map-drawing), if well done will obtain a large number of marks.

1. Draw a map of

(a) England and Wales (showing the principal coal
fields).

Or (b) The Mediterranean sea, its shores and islands.
Or (c) Lower Egypt.

N.B.-If the candidate put in and correctly number the lines of lattitude and longitude it will add to the value of the exercise.

Places must not be indicated by letters or figures, referring to a list of names at the side, but the names themselves must be inserted in the map.

2. Enumerate the different lines which are usually traced on a map of the world; and explain more fully how the lines of latitude and longitude are determined. Why are degrees of longitude not equal in length?

3. Name six of the towns in Great Britain which have greatly increased in size during the last half century. Account

for their rapid growth and describe their chief articles of manufacture.

4. Write a list of the principal lines of railway which radiate from London, and fully describe the course of one of them.

5. Describe in words, with a rough sketch map, a fortnight's summer trip in Scotland, or in North Wales, or in the west of Ireland, giving a short account of the principal objects of interest visited.

6. A workman in London was anxious to proceed to Vienna viâ Antwerp. Sketch the route which he should take, and briefly describe the chief places of importance which he would pass through.

7. Give a description of one of the chief passes over the Alps to Italy.

8. Name the States of America touching the Atlantic and Gulf of Mexico, with their extent, produce, population, and claims to importance.

9. From which of our possessions do we import tea, coffee, sugar, sago, wool, tin, gold, and indigo? Give a short account of the manner of obtaining three of these articles.

10. In Africa, name the states dependent upon Turkey, European Possessions, and Native or Independent States. Write a short history of one of the first and last groups.

11. Describe the North coast of North America with especial reference to the tracks of recent Arctic voyagers.

12. Write full notes, without any introduction, on the valley of the Nile.

13. Name the five principal races of the world, the countries which they respectively inhabit and their systems of religion. Give the general character of each race.

14. Where are the following places, and for what are they remarkable :-La Plata, Savannah, Venice, Sydney, Hong Kong, Lima, Riga, Hamburgh, Gondar, Christiana, Utrecht, Tripoli, Pernambuco, and Fernando Po?

MALE CANDIDATES.-SECOND YEAR.

ALGEBRA AND MENSURATION.

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

The solution must be given at such length as to be intelligible to the Examiner, otherwise the answer will be considered of no value.

[NOTE. In all problems, where required, the circumference of a circle may be assumed to be =(22)ths of the diameter. Not more than 2 decimal places are required in the answer.]

ALGEBRA.

1. Find an expression for the remainder when aa12 + ɑn-¡Ã1-1 + &c. + α1x + α, is divided by x −

- p.

Hence prove that any number in the denary scale is divisible by 11, if the difference between the respective sums of the odd and even digits is divisible by 11.

2. Explain the reason for marking off pairs of figures in finding the square root of a number.

Find the square root of 9064314·49 and of

4. When is one quantity said to vary as another?

The pace of a tortoise on a grass plot varies inversely as the length of the grass. If it can travel 6 feet a minute when the grass is an inch long, what is the length of the grass when it takes 15 minutes to travel 20 yards? 5. Find the number of combinations of n things r together.

How many things are contained in each combination of the set which gives the greatest possible number of combinations?

Prove that the greatest possible number of combinations of 2n things is double the greatest possible number of 1 things.

2n

6. Write down the first three terms and the (r + 1)th term of the

expansion of (1 + n).

m n

If p be the integral part of the fraction prove that the

first p+1 terms of the expansion are positive, and that the succeeding terms are alternately negative and positive. 7. Find the amount of £P in n years at r per cent. per annum compound interest.

In what time will a sum of money double itself at 21 per cent.?

Log 1·025·01072.log 2 = ·30103.

8. Define the logarithm of a number to a given base, and explain the terms "mantissa" and "characteristic."

Prove that loga × log b = 1. What use is made of this theorem.

9. The nth term of a series is 2n + 2", find the first three terms and the sum of n terms.

10. A train performs one-third of a journey of 243 miles at a certain rate, and the rest at 9 miles an hour less, thus completing the journey in 36 minutes longer than it would have, had the original speed been continued throughout. What time is now required for the whole journey?

MENSURATION.

1. The weights of two similar pyramids are as 49: 36 and their densities: 67; find the ratio of their heights.

2. The area of the surface of a sphere is 154 square inches, find its volume.

3. Find the volume of a frustum of a cone of length 1, the diameters of the two ends being a and b respectively.