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Translate the following passage into Greek Verse :
Now, most noble Brutus,
Bru. Even by the rule of that philosophy,
For fear of what might fall, so to prevent
Translate the following passage into Latin Lyric Verse:
Life is but a day at most,
Translate the following passage into Latin Prose :
Valerian was about sixty years of age when he was invested with the purple, not by the caprice of the populace, or the clamours of the army, but by the unanimous voice of the Roman world. In his gradual ascent through the honours of the state, he had deserved the favour of virtuous princes, and had declared himself the enemy of tyrants. His noble birth, his mild but unblemished manners, his learning, prudence, and experience, were revered by the senate and people; and if mankind (according to the observation of an ancient writer) had been left at liberty to choose a master, their choice would most assuredly have fallen on Valerian. Perhaps the merit of this emperor was inadequate to his reputation; perhaps his abilities, or at least his spirit, were affected by the languor and coldness of old age. The consciousness of his decline engaged him to share the throne with a younger and more active associate: the emergency of the time demanded a general no less than a prince; and the experience of the Roman censor might have directed him where to bestow the imperial purple, as the reward of military merit. But instead of making a judicious choice, which would have confirmed his reign and endeared his memory, Valerian, consulting only the dictates of affection or vanity, immediately invested with the supreme honours his son Galienus, a youth whose effeminate vices had been hitherto concealed by the obscurity of a private station.-GIBBON.
As youth and love, with sprightly dance,
1. The three points are in a right line in which each side of a triangle inscribed in a circle is met by the tangent at the opposite vertex?
2. The rectangle under the perpendiculars from any point of a circle on two tangents is equal to the square of the perpendicular on their chord
3. If a line be cut harmonically, any circle through one pair of conjugate points is cut orthogonally by the circle whose diameter is the distance between the other pair of conjugates?
4. The circles whose diameters are the three diagonals of a complete quadrilateral have the same radical axis ?
5. If a line be cut in extreme and mean ratio, and the greater segment divided again in extreme and mean ratio, and so on; find the sum of all the greater segments.
6. Describe a circle through a given point to touch a given line and a given circle; how many such circles can be described ?
7. A tangent is drawn to a given circle whose centre is O, and its pole is taken relatively to another circle whose centre is C; prove that the dis
tance of this pole from C is to its distance from the polar of O, the centre of the former circle, with regard to the circle whose centre is C, in a constant ratio.
8. Find the locus of a point from which tangents to two fixed circles shall have a given ratio, and prove the principle upon which the solution depends.
9. In every hexagon inscribed in a circle the two triangles determined by the two sets of alternate sides are in perspective, opposite sides in the hexagon being corresponding sides in the perspective; prove this, and demonstrate the principle upon which the proof is based.
[Triangles are said to be in perspective when lines joining corresponding vertices meet in a point, their vertices and sides corresponding in pairs.]
10. When three lines through the vertices of a triangle meet in a point, their three points of intersection with the opposite sides determine an inscribed triangle such that the intersections of its sides with the corresponding sides of the original triangle lie in a straight line.
II. Prove that the circle which passes through the feet of the perpendiculars from the angles on the opposite sides of a triangle bisects the distances between the vertices and the point of intersection of these perpendiculars.
12. Two non-intersecting circles have their centres at A and B, and the line joining AB is cut at I so that AI2 - BI2 = the difference of the squares of the radii AR and BS; two points E and F are found so that IE2= IF2 = AI2 - AR2 = BI2 — BS2. Prove that any circle through E and F will cut these circles at right angles, and show that
13. The intersection of the perpendiculars of a triangle is the radical centre of the circles of which the sides are diameters ?
14. A right line is cut harmonically by two circles; prove that the rectangle under the perpendiculars let fall on it from the centres of the circles is constant.
15. The bases of two triangles inscribed in a circle are fixed; prove that the line joining the intersections of corresponding side passes through a fixed point.
16. Two points can be found which have the same polars with respect to two circles; and these points are harmonic conjugates with the centres of similitude?
17. Find the locus of a point at which two portions of the same line subtend equal angles.
18. If a quadrilateral be inscribed in a circle, and another circumscribed touching at the angular points, prove that their third diagonals are coincident, and that the intersection of each pair of their three diagonals is the pole of the remaining one.
1. Expand by the Binomial Theorem (a+b), and apply the result to calculate the cube root of 100 to four places of decimals.
2. Solve the system of equations,
(1 + x) (1 + y) = 10, x2y + y2x = 18.
3. A bankrupt has three creditors, of whom A receives a shilling in the pound more than B, and B a shilling in the pound more than C; A is thus paid £100 less than B, and B £100 less than C; the total debts were £3000, and the assets £1200; how much was due to each?
4. Given a polygon of n sides, with no re-entrant angle; how many pairs of diagonals intersect inside the figure, and how many outside? 5. Prove that
vanishes when n is odd; and that when n is even it is equal to the coefficient of the middle term in the expansion of (x − y)”.
(2 − x)2 (1 + x) series of ascending powers of x.
7. Solve the simultaneous equations
by the Binomial Theorem.
9. The sum of £700 was divided among four persons whose shares were in geometrical progression, and the difference between the greatest and least was to the difference between the means as 37 to 12; what were the respective shares ?
10. Eliminate x between the equations
X 27 + =
into partial fractions, and expand it in a
11. If ax+by+cz=d, a'x+by+c'z=d', a′′x+by+c"z=d", write down the values of x, y, z.
12. Solve the equations
x (y + z) = α, y(x+z) = b, z(x+y) =
13. If y = ax + bx2 + cx3 + &c., show how to express x in terms of y; and hence deduce the series for a from that for log (1 + x).
14. Resolve into its partial fractions
1+2x+3x2 + 4x3
(1 + x)2(1+x+x2)'
15. Explain the method of finding the sum of a recurring series; and find the sum and general term of the series
1-3x+5x2-7x3 + 9x1 — &c.
1. The sides of a triangle are 13, 14, 15; find the tangents of the three angles.
2. Prove that the segments of the perpendicular of a triangle are 2R cos C and 2R cos 4 cos B.
3. When m and n are integers, prove that am divided by an is am-n ; and show that
By actually raising 6 to its powers, show that the common logarithm of 6 is a little more than 3.
I a-n =
4. Given in a triangle base, vertical angle, and sum of cotangents of base angles; how would you calculate the sides?
5. Given four sides of a quadrilateral, and the angle at which two opposite sides intersect; give a process adapted to logarithms for calculating the angles of the figure. Of how many solutions does the problem admit?
7. If 2 cos 0=:
6. Prove that the sides of the triangle formed by joining the centres of the three exscribed circles are
4R cos A, 4R cos B, 4R cos C, where R is the radius of the circumscribing circle.
2 cos (0-4)
2 cos (0 +4)=xy +;
;, prove that