3. Translate into French

Nature has given us two ears and one mouth, in order that we should hear much and say little.

Regard as a true friend the sincere man who apprises you of your faults; not him who approves of all you say and all you do.

When I see the birds forming their nests with so much art, I ask what master has taught them mathematics and architecture.

4. Give the gender and meaning of the following words :--

6. Translate the following letter into French and put a proper beginning and close to it.

We had beautiful weather for the agricultural show at Newton.

The number and variety of animals gave me a good opportunity of executing your commission. I bought a Devon bull and a lot of Leicester sheep for you, which I hope will please you. I shall despatch them by the steamer from Southampton next week.

The great attraction to visitors was the Steam Plough, which succeeded admirably, performing seven acres a-day with ease.

DEPARTMENT C.

ELEMENTARY PAPER.--(THURSDAY MORNING.)

1. Add nine thousand and four to twenty-three thousand seven hundred and five, and subtract the answer from 40,000.

2. If an orchard contain 78 apple-trees, and each tree produce on an average eleven hundred and six apples, what is the whole number of apples in the orchard?

3. A dealer buys 157 head of cattle for 20147.; how much a head do they cost him?

4. Find the cost of 2247 articles, at 17. 16s. 44d. a piece.

5. A farm consists of 23 fields, of which 7 measure, on the average, 13 a. 3 r. 17 p. apiece; 8 others measure 9 a. 0 r. 35 p. each; and the rest 6 a. 1 r. 8 p. each: what is the acreage of the farm?

6. Reckoning each furrow as 9 inches in breadth, how far will a ploughman walk in ploughing an acre of land?

7. Calculate the cost of wintering a bullock in a straw-yard for 24 weeks, allowing him 24 lb. of oil-cake daily, and accepting the manure made as an equivalent for the straw consumed, the cake being assumed to cost 117. per ton.

8. A bankrupt's debts are 49761., and he is able to pay 11s. 8d. in the pound: what are his effects worth?

9. A gentleman has a poor meadow that produces him annually about ton of hay per acre, and he is advised to convert it into a water-meadow, at a cost of 12l. 10s. per acre: what must be the produce of hay per acre, after the improvement, in order that it may pay the owner ten per cent. upon his outlay; supposing the hay to be worth 31. per ton on an average of years?

10. If I give 371. for a load of hay weighing 17, cwt., what should I pay for a whole rick weighing 27 tons?

11. If 7 horses carry 80 tons of manure a distance of 1 mile in 3 days of 10 hours each, how many days of 7 hours each will be occupied in carting a heap of 29 tons to a distance of 725 yards with 4 horses?

12. Calculate the rates on a rental of 2637. 10s. at 2s. 3d. in the pound, the rateable value being two-thirds of the actual value.

13. In a farm of 384 acres, gths are arable, 17ths meadow, and the rest garden and orcharding: how many acres are there of each sort?

14. If railway shares be bought at 911. 5s. per share, how must they be sold in order to gain exactly 4 per cent. by the transaction?

15. If the diameter of a penny-piece be 1.35 in., how many pennypieces laid together will cover a distance of 100 yards?

16. In a parish of 725 people, 24 are children, 36 are men or boy's, and the rest women or girls: how many are there of each class?

17. If the toll at a gate for one horse be 3d. and the toll be altered to 12 mils (1000 mils to the £), what will be the loss per cent, on the revenue derived from the horse-toll?

1

2c, 6b -3a

7c, and 5a 4b+ 5c;

18. Add together 4a 36
and subtract the result from a + b + c.

19. Multiply 52 - 6xy + y2 by 5y + 6xy + x3.

Find the co-efficient of x3 in cx + d by x2

2a2 b2 + b by a + b.

3

ax + bx2

ax + b.

23. When is one quantity said to be a measure of another? Show how to find the greatest common measure of 2 algebraical quantities, and extend the rule to 3 or more quantities.

Ex. Find the greatest common measure of x2 - 2x 3, x2 - 7x+ 12, and x2 - 6.

2m

25. Cube 4a2bc-2, and extract the square root of the result.

28. At present A is just 3 times as old as his son B, but 16 years hence he will be only twice as old: what are the ages of A and B ?

29. A labourer, having done a piece of work for a certain sum of money, finds that if he had done it in 2 days less he would have earned 6d. a day more; and if he had taken 3 days more to do it, his daily earnings would have been 6d. less in what time did he do it, and what were his earnings?

30. Show how to describe an equilateral triangle upon a given finite straight line.

What figure would be formed by placing two equilateral triangles base to base?

31. Prove that any two sides of a triangle are together greater than the third side.

Is the same proposition true of the angles of a triangle? Give reasons for your answer.

32. Prove geometrically that, 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.

33. Prove the last proposition algebraically also.

34. If a straight line touch a circle, the straight line drawn from the centre to the point of contact shall be perpendicular to the line touching the circle.

35. What is meant by the angle in a segment of a circle?

Prove that the angle in a semicircle is always a right angle.

36. Apply the last proposition to construct an oblong, of which the diagonal and one of the sides are given.

37. Give the practical method of testing whether an angle is a right angle or not.

C.

PURE MATHEMATICS.-(THURSDAY AFTERNOON.)

1. Extract the cube root of 1157.625.

2. Show that the product of any 3 consecutive numbers is divisible by 6.

3. What is a prime number? Show how to determine whether any given number is a prime number or not.

4. What year of the Christian era would this be, if six were the radix of our scale of notation instead of ten? On what scale would the present year be represented by 83v where v represents twelve?

5. Find the side of a square containing, in superficial measure, 763 feet 18 inches and 30 seconds.

6. Find the vulgar fraction equivalent to the circulating decimal 0.725656.

7. What is meant by incommensurable magnitudes? Explain the term with reference (1) to the side of a square and its diagonal, (2) with reference to the circumference and diameter of a circle.

8. Define ratio and proportion, and show that whenever 4 numbers are in proportion the product of the two extreme terms is equal to the product of the two mean terms. What use is made of this property in practical arithmetic?

10. Enunciate the binomial theorem, and prove that the nth term

12. Find two numbers such that they shall bear to each other the ratio of 3 to 4, and their sum shall bear to the sum of their squares the ratio of 14 to 1.

13. Define a logarithm, and prove that the sum of the logarithms of two numbers is the logarithm of their product.

14. Having given log. 2 = 0.30103, find the logarithms of

and

of 25.6.