2. ABCD is a circle, AD being a diameter, and AC, DB two chords intersecting in O; prove

that the sum of the rectangles AC, AO and DB, DO is constant.

3. A straight line is said to be cut harmonically when it is divided internally into two parts and externally into two parts, so that the ratio of the former is equal to that of the latter. Show how to divide a straight line harmonically, and prove that if one circle divides the diameter of another harmonically the two circles cut one another at right angles.

4. Prove that the area of any rectilineal figure described on the hypotenuse of a right-angled triangle is equal to the sum of the similar figures described on the sides.

5. Through a point O in a parallelogram straight lines MOM', NON' are drawn parallel to the sides cutting them in M, M', N, N', MN and M'N' intersect in P; find the locus traced by P as O moves.

6. Construct a quadrilateral not symmetrical with respect to either diagonal, so that it can have a circle inscribed in it and a circle circumscribed about it, and show that in relation to the same two circles an infinite number of such quadrilaterals can be described.

7. Trace the changes in the value of

sin 0 cos 0

sin

cos 0

as 0 varies from o to π.

8. Any point is taken in a triangle PQR and joined to the angular points by lines PO, QO, and RO meeting the sides in points L, M, and N; prove that

sin PON sin QOL sin ROM= sin QON sin ROL sin POM.

9. Show that if a quadrilateral whose sides are a, b, c, d, is inscribed in one circle and circumscribed about another, its area is √abcd.

10. If r, r1 r2

r3

are the radii of the inscribed and escribed circles of a triangle, prove that

11. Two straight lines, including an angle y, intersect at O, and points A and B are taken one on each line at distances a and b from 0; it is desired to join these by a curve composed of two arcs of equal circles touching one another at their junction and touching the straight lines at the points A and B; obtain an equation for the length of their radius, and interpret its two roots.

ENGLISH.

The Board of Examiners.

1. Write an essay on the following thesis :-All that raises us above the level of the savage comes from the shores of the Mediterranean.

2. Analyse the following passages :

(a)

Or turning to the Vatican, go see
Laocoon's torture dignifying pain—
A father's love and mortal's agony

With an immortal's patience blending.

(b) That he acted thus was a misfortune because although such conscientious thoroughness is in itself admirable unless the historical knowledge so acquired be rigidly subordinated to the purposes of art it can only lead the artist into

error.

3. Parse the words following in italic :—

To be, or not to be,-that is the question:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles,
And by opposing end them?

4. Begin at each of the following openings, and complete four lines :—

5. What is the meaning of each of the following words? Have they undergone any change in meaning? Trace their origin :-Baffled, chimerical, conceit, dungeon, factor, fantastic, gazette, hind, inspiration, mansion.

6. Originally the words in italic stood in the following passages:

(a) And leaves the bonny traitor in the Tow'r.

(b) Toil, envy, want, the garret and the gaol. (c) Some village Cato, that with dauntless breast The little tyrant of his fields withstood, Some mute inglorious Tully here may rest, Some Caesar, guiltless of his country's blood. Comment on the significance of each change.

7. Explain the following passages :—

(a) The poet that beautified the sect that was otherwise inferior to the rest,

(b) The astrologers call the evil influences

(c) Mountebanks for the natural body.

(d) When any of the four pillars of government are mainly shaken,

(e) Iterations are commonly loss of time; but there is no such gain of time as to iterate often the state of the question.

(f) Speech was like cloth of Arras.

8. The following words are from Macbeth:-Paddock, minion, sennight, thane, limbec, sleave, shard, seeling, newt, foison. Write a note on each.

9. Contrast Lady Macbeth at the beginning and at the end of the play.

10. Name Bacon's principal works. What objection is there to the name "Lord Bacon "?

11. Specify the circumstances in which the relative that is preferable to the other relatives, and give examples.

12. Write a full account of Humour, the origin of the word, and its changes of meaning.

HISTORY.

The Board of Examiners.

Write careful Essays upon SIX, and only Six, of the following subjects.

1. The course of events which brought about the submission of King John to the Papal legate.

2. The origin, development, and functions of the mediæval guilds in England.

3. The history of the war with France in the reign of Henry the Sixth.

4. The origin and the results of the trial of " the Seven Bishops."

5. The influence of the French Revolution upon English literature and English legislation.

6. The origin, development, and functions of the Roman Comitia.

7. The influence of Greece upon Roman religion and upon Roman literature.

8. The permanent influence of Persia on the history of Greece.

9. Grecian history as illustrating the dangers arising from exaggerated party spirit.