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2. What is the axiom on which Euclid bases his reasonings on parallel lines P Is Proposition 17 of the First Book the converse of that axiom? If so, is there any objection to the axiom ?

If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side; and also the two interior angles on the same side together equal to two right angles.

3. In any right-angled triangle, the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.

From the middle point of a side of a right-angled triangle a perpendicular is drawn to the hypotenuse; show that the difference of the squares on the segments into which it is divided is equal to the square on the other side.

4. By what proposition of the First Book is it proved that the area of a triangle whose altitude is a units long, and whose base is b units long, is į a b ?

If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

5. Why cannot we satisfactorily demonstrate propositions of the Second Book by algebraical processes ?

In every triangle, the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle.

ALGEBRA.

6. Express algebraically :-The fourth power of tho sum of two numbers (a and b), together with twice tho product of their squares is equal to the sum of their fourth powers together with four times the product of their product and the square of their sum. Also verify it, when a=

=2,b=3.

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7. Subtract:(x+y) (3a - 26) from (x+y) (3a+26); and divide wa+y+ 1-2y+2x – 2xy by 2 - Y+1.

8. *Prove the rule for dividing one algebraical fraction by another, the letters denoting any numbers.

23 Simplify (x + 2 + 9. Solve the equations:

7+ 3x 1. – 17( =(

8 2. <28: +8°4=369.

32x8ly.

30 3.

6

=X - 9. 10. Prove that the difference between the cube of the sum of any two numbers and the sum of their cubes is divisible by three times their product.

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EUCLID. 11. To draw a straight line from a given point without the circumference, which shall touch a given circle.

Show that two such lines can be drawn, and that they are equally inclined to the line which joins the given point with the centre of the given circle.

12. If a straight line touch a circle and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle. Write out the converse of this proposition.

MENSURATION. 13. The area of a chess-board which contains 64 equal squares and an outer rim an inch wide, is 134:56 squaro inches : find the side of each square. Find also the width of the outer rim of another board of the same size, in which the area of each square is 1.360 square inches.

14. The three sides of a triangle are in the ratio 4: 5: 7, and their sum equals 32 yards, find the area of tho triangle in yards, to two places of decimals.

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DOMESTIC ECONOMY.

Three hours allowed for this paper. Candidates are not permitted to answer more than one Question in each Section.

SECTION I. (Needlework.)-1. Describe fully the following stitches, and say in what garments and materials they are commonly used, and how you would teach them to children :-hemming, felling, back-stitching, featherstitching (i.e., coral stitching).

2. Describe the process of cutting out and making a pinafore for a girl of six years old, with exact measurements, and an account of the materials required.

SECTION II. (Savings and Investments.)-1. State the annual income on which, in your opinion, a retired Schoolmistress could live in comfort in her old age; and the methods of saving and investment in her days of full work and salary, by which she could provide that income on retirement.

On what weekly wages can an artisan maintain in comfort himself, his wife, and four children between the ages of four and ten :-and in what proportion should he distribute those wages in rent, food, clothing, education of his children, recreation, and savings?

Section III. (Food--its ingredients.) – 1. What effect is produced on the human body by food containing in large quantities the following substances respectively: (a) sugar, (b) lime, (c) salt, (d) animal oils ?

2. State fully what are the objections to a diet either exclusively vegetable or exclusively animal.

SECTION IV. (Foodits preparation.)-1. Describe the efficient modes of cooking potatoes ; give your own opin. ion as to the merits of each, and say for what dish of meat each mode of cooking them is most suitable.

2. Give an account of the materials, preparation, and cooking of an economical and wholesome dinner for a Schoolmistress living alone.

Section V. (Rules for Health.)-1. Mention any res. pects in which the modern fashion of female dress is injurious to health, and show in what way each foolish practice in dressing produccs its bad effects.

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2. Give plain rules for the preservation of health for a Papil Teacher (a) who lives in the country a mile from her school, and (b) for a Pupil Teacher in London living a few doors from her school.

3. What would you do before the doctor came, if a child in your school (a) was badly scalded, (b) had fainted, (c) had cut his arm above the elbow ?

SECTION VI. (Clothing and Washing.)-1. Describe the modes of washiug, drying, and getting up the different articles which would go into the tub on washing-day in a labourer's cottage.

2. Give an account of the price, material, colour, and making up of a neat dress for your own summer wear in school, and say how it should be washed and worn so as to last as long as possible,

FEMALE CANDIDATES.

ARITHMETIC. Three hours allowed for this paper. Candidates are not permitted to answer more than one Question in each Section.

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.-1. The first of 4 parcels of money contained two hundred and six pounds, and twopence; the second fifty sovereigns, seventeen half-sovereigns, and nine half-pence; the third twenty-seven half-guineas, and eightpence; the fourth nineteen half-sovereigns, and three half-crowns. Distribute the amount equally among 29 societies.

2. One room contains 18 sq. yds., 3 sq. ft., 19 in.; a second, 42 sq. yds., 8 ft., 11 in.; a third, 29 sq. yds., 5ft. 100 in. What must be added or subtracted in each case to make the rooms of average size?

Section II.-A silversmith made a certain number of teaspoons weighing 26 lbs. 10 oz. 13 dwt., and a certain number of tablespoons weighing 38 lbs. 10 oz. 11 dwts. 18 grs.; find the cost of all the spoons at £3 178. 11d. State the different methods employed in subtraction, and give your reasons for preferring one of them to another.

per oz.

2. A farmer rents a farm of 400 acres on the following terms :-He pays as rent 100 qrs. of wheat, 75 qrs. of barley, and 60 qrs. of oats, the price of wheat, barley, and oats being respectively 49s. 6d., 30s. 8d., and 198. 20. per quarter. Give his average rent per acre in £ s. d.

SECTION III.-1. Make the following bill:-5 tons of coals at 158. 6d. per ton, carriage of same at 28. 6d. per ton; 2 trucks of gravel (i.e., 11} tons) at 58. per ton, carriage of same at 2s. 6d. per ton.

What are the two different kinds of practice called ? Which kind is mostly used in bills of parcels P Give reasons for your answer.

9. Find the change out of a £10 note after paying the following bill :—12; yds. of flannel at 1s. 6d. per yd., 37 yds. of calico at 1s. O d. per yd., 21 yds. of muslin at 28. 4 d. per yd., 18 yds. of linen at 28. 6d. per yd.

What is a Bill of Parcels ? What else is it sometimes called ? What rules of mental arithmetic can be applied in finding the amounts of the items ?

SECTION IV.–1. Find by practice the rent of 311 acres 2 roods 26 perches at 58. 81d. per rood.

What rule does practice depend upon ? And how is that rule simplified by it?

2. A bankrupt can pay only 12s. 6d. in the pound, and his debts amount to £1,537 4s. 4d.; what is his estate worth P How much will be paid on a debt of £276 118. 6d.

What is meant by an “aliquotpart? SECTION V.-1. The planting of a rood of ground cost £28 8s. 4d.; what was paid for planting 23 acres 3 roods 24 perches and 11 sq. yds. ?

Explain and define measure, factor, multiple, submultiple.

2. Bought 176 yds. 2 qrs. 2 nails, 17 inch, at 18s. Od. per English ell (= 5 quarters); what is the gain or loss in selling at 6d.

per

inch ? Explain the terms profit and loss--profit and loss per cent.

SECTION VI.-1. If the road in front of a row of

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