1. Construct a triangle equal in area to a convex fivesided rectilineal figure ABCDE; AB is to be

the base of the triangle, and AE the direction of one of its sides.

2. The area of any two parallelograms described on two sides of a triangle is equal to that of a parallelogram on the base whose side is equal and parallel to the line drawn from the vertex of the triangle to the intersection of the two sides of the former parallelograms produced to meet.

3. If from any point straight lines be drawn to the angles of a rectangle the sums of the squares on those drawn to opposite angles are equal.

4. The circle which passes through the middle points of the sides of a triangle passes also through the feet of the perpendiculars on the sides from the opposite angles, and also bisects any straight line drawn from the intersection of these perpendiculars to the circumference of the circumscribing

circle.

5. If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the rectangle contained by the sides of the triangle is equal to that contained by the segments of the base together with the square on the straight line which bisects the angle.

Prove that this rectangle is also equal to that contained by two straight lines equally inclined to the bisector, one terminated by the base and the other by the circumscribing circle.

6. If four straight lines are proportionals, the rectangle contained by the extremes is equal to that contained by the means.

If the exterior angle of a triangle be bisected by a straight line which cuts the base produced, the square on the bisecting line is equal to the difference of the rectangles of the segments of the base and of the sides of the triangle.

7. Define the cosine of an angle. Shew how to resolve the sum or the difference of two cosines into factors.

If A + B + C = nπ, prove that

cos 2 A+ cos 2 B+ cos 2C+1
= (− 1)" 4 cos A. cos B. cos C.

8. Define the secant and the cosecant of an angle, and shew that they are the reciprocals of the cosine and the sine of the angle respectively.

If

sec B√3. sec A

and cosec 2 B = cosec 2 A,

find A and B in the most general form.

10. If 01 02 03 04 are so nearly equal that the cosine of the difference of any two of them may be

assumed to be unity, prove that the last equation in question 9 becomes

11. Shew how to solve a triangle, having given two sides and the included angle. If the third side only is wanted, shew how it may be obtained without first finding the other angles.

If C

36° 52′ 12", a 70, 35, find = b = A and B, having given log 304771213, and L cot (18° 26' 6")

10-4771213.

12. Find the radius of the inscribed and of any escribed circle of a triangle in terms of its sides. If A be the area of the triangle whose base is the side a of another triangle and vertex the centre of the inscribed circle of the latter, and A', the area of another triangle having the same base and vertex the centre of the escribed circle which touches a, with similar notation for the other sides, prove

that

A,

A2

A3

+

+

= 1.

A'2 A's

ENGLISH.

The Board of Examiners.

1. (a) In the following passages parse fully each word which is printed in italics.

(b) Write a note on any peculiarities of construction which you may remark in them.

D

Tullie when he was to drive out Catiline.

Which is not to take upon me to teach Poets
how they should do, but onely finding my
selfe sick among the rest, to shewe some
one or two spots of the common infection,
growne among the most part of writers:
that acknowledging our selves somewhat
awry, we may bend to the right use both
of matter and manner.

Which did refuse three thousand ducats of me
And begged the ring.

-here in her hairs

The painter plays the spider.

For you shall hence upon your wedding-day.
What mercy can you render him, Antonio?

2. Make a full analysis of each of the following passages:(a) There is again another serious difficulty in the way of an enlightened choice of representatives by a popular constituency, a difficulty, it may be observed, which extends to every mode of selection, and equally besets the choice of members of a house of peers by the Crown.

(b) Not sedentary all: there are who roam

(c)

To scatter seeds of life on barbarous shores.

Nowe, wherein we want desert, were a thankeworthy labour to expresse: but if I knew, I should have mended myselfe.

(d) Doth not knowledge of law, whose end is to even and right all things, being abused, grow the crooked fosterer of horrible injuries?

3. Write a short note on the philology of each of the following words, and give its meaning:

4. (a) What is the rule as to the position of prepositions in a sentence? and what are the exceptions?

(b) What do you mean by a pronoun being a relational word? What other words are relational? Give examples.

(c) By what may adjectives be replaced in a sentence?

5. Write out three stanzas of the Hymn on the Nativity, commencing-" Such musick as 'tis said," to the peering day."

66

6. (a) Write a very short account of Keats, his works, and friends.

(b) Who is referred to as the sixth phantom in each of the two groups described in the Vision of Public Credit?

(c) What does Addison consider to be the great danger in giving any man absolute power? (d) "The food often grows in one country, and the sauce in another." How does Addison illustrate this?

(e) What are the things which De Quincey says that woman cannot execute so well as man, and in what does he consider that she can surpass him, taking Joan of Arc as an example?