Translate into Greek Iambics:

Dar'st thou denie this? wherefore art thou silent?
Speake, and this shall convince thee. Here they are,
I see 'hem, in this Senate, that were with thee.
O you immortall Gods! in what clime are wee?
What region do we live in? in what ayre?
What common-wealth, or state is this we have?
Here, here, amongst us, our owne number, Fathers,
In this most holy Councell of the world,

They are, that seeke the spoyle of me, of you,
Of ours, of all; what I can name's too narrow:
Follow the Sunne, and find not their ambition.
There I behold, being Consull; Nay, I aske
Their counsels of the State, as from good Patriots:
Whom it were fit the axe should hew in pieces,
I not so much as wound, yet, with my voyce.

2. From Atqui ne ex eo quidem tempore.... .ferrum nota recepit?

to

The Druids were accustomed to dwell at a distance from the profane, in huts or caverns, amid the silence and gloom of the forest. There, at the hours of noon and midnight, when the Deity

was supposed to honour the sacred spot with his presence, the trembling votary was admitted within a circle of lofty oaks, to prefer his prayer, and listen to the responses of the minister. In peace they offered the fruits of the earth: in war they devoted to the god of battles the spoils of the enemy. The cattle were slaughtered in his honour; and a pile formed of the rest of the booty was consecrated as a monument of his powerful assistance. But in the hour of danger or distress human sacrifices were deemed the most efficacious. Impelled by a superstition, which steeled all the feelings of humanity, the officiating priest plunged his dagger into the breast of his victim, whether captive or malefactor; and rom the rapidity with which the blood issued from the wound, and the convulsions in which the sufferer expired, announced the future happiness or calamity of his Country.

For Latin Hexameters :

LINGARD. Hist. of Eng., ch. 1.

And ye, Pierian Sisters, sprung from Jove
And sage Mnemosyne,-full long debarred
From your first mansions,—exiled all too long
From many a hallowed stream and grove,
Dear native regions where ye wont to rove,
Chanting for patriot heroes the reward
Of never-dying song!

Now (for, though Truth descending from above
The Olympian summit hath destroyed for aye
Your kindred Deities, ye live and move,
And exercise unblamed a generous sway)
Now, on the margin of some spotless fountain,
Or top serene of unmolested mountain,
Strike audibly the noblest of your lyres,
And for a moment meet my soul's desires!
That I, or some more favoured Bard, may hear
What ye, celestial Maids! have often sung
Of Britain's acts,-may catch it with rapt ear,
And give the treasure to our British tongue!

WORDSWORTH. Ode, Jan. 1816.

V.-EUCLID AND GEOMETRICAL CONICS.

1. Define a plane angle, a right angle, a circle and a rhomboid.

2. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is, divides them into two equal parts.

Prove also that the two diameters bisect each other.

3. Divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part

Solve the same problem algebraically, and give the geometrical interpretation of the result.

4. The opposite angles of any quadrilateral figure described in a circle, are together equal to two right angles.

If two opposite sides are equal the other two are parallel. 5. Describe a circle touching one side of a triangle and the other two sides produced.

Also describe a circle touching three sides of a parallelogram. 6. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangle shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

If two straight lines be drawn through the same point 4, one touching a circle in B and the other cutting it in C and D, the triangles ABC, ABD are similar.

7. If two straight lines be parallel, and one of them is at right angles to a plane; the other also shall be at right angles to the same plane.

8. Prove that in the parabola, SY is a mean proportional between SP and SA.

If a perpendicular SZ be let fall from S on the normal at P, the locus of Z will be another parabola of which S is the vertex.

9. If NP be the ordinate of a point P in an ellipse whose centre is C, and the tangent at P`meet the axis produced in T, prove that CN. CT = CA.

Prove also that the subnormal is a third proportional to CT and BC.

10. If CP and CD are semi-diameters in any ellipse and CD be parallel to the tangent at P, CP is also parallel to the tangent at D.

Shew also that the distance of D from the axis major bears a constant ratio to the distance of P from the axis.minor.

11. Define an asymptote to a curve, and shew that the diagonals of the rectangle formed by drawing tangents at the vertices of an hyperbola and its conjugate are asymptotes to both curves.

VI.-ALGEBRA AND TRIGONOMETRY.

1. Find the amount of £62. 10s. in four years at 20 per cent. per annum compound interest.

2. Shew that when n is integral

an integer.

n. n + 1| 2n + 1]

is also

6

Find the forms of n for which this integer will be even. 3. Extract the square root of

(m + n)2 − 4 (m − n) √(mn).

What condition must numerical values of m and n satisfy that the root may be rational?

4. A contractor engages to execute work in n days. After m days, fearing not to finish in time, he doubles the number of his men and thus finishes p days before the specified time. Find whether or not he would have failed in his contract without increase of labourers.

5. What do you understand by an inductive proof?

Shew by such proof that the sum of the first n odd numbers is no.

Prove that the continued product of the first n odd numbers is equal to

2n . (2n − 1)...(n + 1)

2"

a - x

6. Shew that all the terms of the expansion of a - x in ascending powers of x are positive.

Approximate to √(2) by the form (1 − 1)*.

7. If P be the nth convergent to a +

In

1

1

shew that

P1 = apn-1+P-2 and ¶„ = Pπ-1*

8. Shew that cos A - sin A = √(2) . sin(45° – A).

Trace the changes in sinA + cos A, as A changes through the first four quadrants.

9. If sin 20 = 1, find all the values for sine.

Indicate by a figure the angles whose sines must be given by these values.

10. Find the radius of the circle circumscribing a triangle in terms of the elements of the triangle.

Shew that the smaller circle in Euc. IV. 10, is equal to the circle circumscribed about the constructed isosceles triangle.

11. Obtain sine in terms of sines of multiples of 0.
Deduce sin' in terms of cosines of multiples of 0.
12. Sum the series

VII.—ALGEBRAICAL GEOMETRY.

1. Find the equation to a straight line which passes through a given point (hk) and makes a given angle (a) with the line.

2. Shew that the equation 3y2 - 8xy - 3x2 = 0 represents two straight lines, and find the angle between them.

3. Investigate the condition necessary that three straight lines whose equations are u = 0, v = 0, w = O may meet in a point. Ex. The three perpendiculars let fall from the angular points of a triangle on the opposite sides all meet in a point.

4. Shew that two tangents can always be drawn to a circle from an external point (hk), and find the equation to the line joining the two points of contact.

If the point (hk) move along a straight line, all the chords of contact will pass though a fixed point.

5. Investigate the equation to the tangent at any point (x'y') of a parabola, and deduce from it the equation to the normal.

How many normals can be drawn to a parabola from a given point?

6. Find the length of the perpendicular from the center on the tangent at any point of an ellipse and the locus of the points of intersection.

7. Find the condition necessary that a given point (x11) may be the middle point of the chord of an ellipse whose equation is y = mx + c, and hence find the locus of the middle points (1) of a system of parallel chords, (2) of all chords whose length is constant (= 27).

8. Shew that, in the ellipse, if any two parallel tangents be drawn, and the points in which they meet a third tangent be joined with the center, the joining lines will be conjugate diameters.

9. Prove that in an hyperbola a diameter CP and its conjugate do not both meet the curve.

If D be the point in which the conjugate diameter meets the conjugate hyperbola, PD will be bisected by one asymptote and parallel to the other.