1. Write down the Postulates. How do you distinguish between a postulate and an axiom?

2. Define a plane superficies, a right angle, an acute-angled triangle, a trapezium, a rectangle.

3. The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles.

4. If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

5. The three angles of a triangle are together equal to two right angles; and all the angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

6. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them. Prove also that the diameters of a parallelogram bisect each other.

7. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

8. 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 of the line between the points of section, is equal to the square of half the line.

9. Describe a square that shall be equal to a given rectilineal figure.

10. What does Euclid mean by the equality of two triangles, and under what conditions does he prove two triangles to be equal?

V

ENGLAND.

PRELIMINARY GENERAL EXAMINATION.- -Christmas, 1865.

WEDNESDAY, December 20th.-Afternoon, 3 to 4.

ALGEBRA.

Examiner.-W. J. REYNOLDS, ESQ., M.A., F.C.P.

1. If x=-2, and y = -3, find the numerical value of (5x-2y)3 — (5x2 —2y2) + 5 (x−2y)2.

3x — 5y — (4x — 5y) — (4x+5y) — {7x+3y− (5x—2y)}. 5. Prove that (x2+y2) (a2+b2) − (ax-by)2 = (bx+ay)2. 6. Divide 66-19x4y-9a3y2+47x2y3+9xy-1875 by 2a3-3a2y-4y2+3y3.

7. Resolve the expression 12a2x3-19a2x2y+5a2ay2 into its elementary factors.

8. Find the Least Common Multiple of a2bcx, b2cay, c2abz. Find the Greatest Common Measure of the same quantities, and also that of

6x3-17x3y+16xy-6y3 and 9x3-18x2y+14xу2 — 4y3.

9. Extract the Square Root of

4a2x4-12aba3z +24a2bx2x2 +9b2x2z2 —36ab2xz3 +36a2b2z4.

abe 37-4745+

T

[Turn over.

11. Of a certain sum, A first receives £5, and one-fifth of the remainder; B next receives £10, and one-fifth of what is then left; and C receives the balance, viz. £15. Determine the original sum.

12. Thirty-four gold and silver coins (the number of each kind being the same) are placed at random in a row. A is to have one half of this row, B the other half. A's share is found to include seven gold coins, and the value of it is £6. The value of B's share is £6. 15s. Find the value of each gold coin, and of each silver coin.

Dès que Charles fut maître, il donna sa confiance et le maniement des affaires au conseiller Piper, qui fut bientôt son premier ministre, sans en avoir le nom. Peu de jours après il le fit comte; ce qui est une qualité éminente en Suède, et non un vain titre qu'on puisse prendre sans conséquence comme en France. Les premiers temps de l'administration du roi ne donnèrent point de lui des idées favorables: il parut qu'il avait été plus impatient que digne de régner. Il n'avait à la vérité aucune passion dangereuse; mais on ne voyait dans sa conduite que des emportements de jeunesse, et de l'opiniâtreté: il paraissait inappliqué et hautain: les ambassadeurs qui étaient à sa cour le prirent même pour un génie médiocre, et le peignirent tel à leurs maîtres. La Suède avait de lui la même opinion: personne ne connaissait son caractère; il l'ignorait lui-même, lorsque des orages formés tout-à-coup dans le Nord donnèrent à ses talents cachés occasion de se déployer.-VOLTAIRE, Charles XII.

II. Grammatical Questions.

1. Parse the verbs in italics, and give the principal parts of every one of them.

2. What is the difference between-(i.) comte, compte, conte; (ii.) vain, vin, vingt; (iii.) sans, sens, cens, cent, sang; (iv.) temps, tant; (v.) dans, dent; (vi.) cour, cours (subst.), cours (je), court. 3. When is the word first to be translated by premier, and when by unième ?

4. Give the feminine of maitre, conseiller, comte, roi, ambassadeur.

5. When do you translate to know by savoir, and when by connaître? Examples.

:

6. Translate into English the following idiomatic sentences:(i.) Comment allez-vous? (ii.) Cet habit vous va bien. (iii.) Irezvous à pied ou à cheval? (iv.) Il voudrait me faire aller. (v.) Il se laisse aller à la paresse. (vi.) Allons! (vii.) Allons donc !

(viii.) Cela va sans dire.