simulator ac dissimulator: alieni appetens, sui profusus; ardens in cupiditatibus: satis eloquentiæ, sapientiæ parum. Vastus animus immoderata, incredibilia, nimis alta semper cupiebat. 8. Hunc exitum habuit, tribus et septuaginta annis quinque principes prospera fortuna emensus, et alieno imperio felicior quam suo. Vetus in familia nobilitas, magnæ opes; ipse medium ingenium, magis extra vitia quam cum virtutibus. Famæ nec incuriosus nec venditator: pecuniæ alienæ non appetens, suæ parcus, publicæ avarus: amicorum libertorumque, ubi in bonos incidisset, sine reprehensione patiens, si mali forent, usque ad culpam ignarus. Sed claritas natalium et metus temporum obtentui, ut quod segnitia erat, sapientia vocaretur. Major privato visus, dum privatus fuit, et omnium consensu capax imperii, nisi imperasset. 9. Hinc ope barbarica variisque Antonius armis, Victor ab Auroræ populis et litore rubro Ægyptum viresque Orientis et ultima secum Bactra vehit; sequiturque (nefas) Ægyptia conjunx. Translate into Latin Prose: IV. Cicero chose the middle way between the obstinacy of Cato and the indolence of Atticus. He preferred always the readiest road to what was right, if it lay open to him; if not, he took the next that seemed likely to bring him to the same end; and in politics, as in morality, when he could not arrive at the true, he contented himself with the probable. He often compares the statesman to the pilot, whose art consists in managing every turn of the winds, and applying even the most perverse to the progress of his voyage; so as by changing his course, and enlarging his circuit of sailing, to arrive with safety, though later, at his destined port. He declared contention to be no longer prudent than either while it did service, or at least, no hurt; but when faction was grown too strong to be withstood, that it was time to give over fighting: and that nothing was left but to extract some good out of the ill, by mitigating that power by patience which they could not reduce by force, and conciliating it, if possible, to the interests of the state. He made a just distinction between bearing what we cannot help, and approving what we ought to condemn; and submitted therefore, yet never consented, to those usurpations; and when he was forced to comply with them, did it always with a reluctance that he expresses very keenly in his letters to his friends. MIDDLETON'S Life of Cicero. Translate into Latin Elegiacs; Fairest flow'r, all flow'rs excelling Flowers of Eve's embowered dwelling Mark, my Polly, how the roses How the bud its sweets discloses ;- Lilies are, by plain direction, But, dear girl, both flow'rs and beauty Then pursue good sense and duty, COTTON. V. 1. Find the ARITHMETIC AND EUCLID. rental of 5 acres, 3 roods, 27 poles, at £1. 128. 8d. per acre. 2. Reduce 88 and vulgar fraction, and as a decimal. 3. How much cotton, 4 feet wide, at 3d. per square foot," must be given in exchange for 393.7 metres of French silk, of a yard wide, at 4 francs per square metre: 1 being worth 25.15 francs, and 1 metre being 39.37 inches? 4. What is the amount of £20 in 4 years at 5 per cent. compound interest? 5. Which is the greater rate of interest, £7 for the use of £145, or £41 for the use of £91. 5s. for a year? 6. 3 men, 4 women, 5 boys, or 6 girls, can perform a piece of work in 60 days, how long will it take 1 man, 2 women, 3 boys, and 4 girls, working together? 7. The duty on coffee, roasted or unroasted, is 3d. per pound. It loses 18 per cent. in the roasting. If roasted coffee costs 1s. per pound before the duty is paid, what ought unroasted coffee to cost, neglecting the expense of roasting? 8. What is meant by a Definition, an Axiom, a Postulate, a Problem, a Theorem? Derive the words, and give an example of each. What is meant by a Reductio ad absurdum proof? 9. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other. 10. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. From a given point there can be drawn to a straight line only two straight lines equal to one another. 11. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. 12. If a straight line be divided into two equal and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. 13. Describe a square that shall be equal to a given rectilineal figure. Describe a rectangle equal to a given square, and having its adjacent sides together equal to a given straight line. 14. If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the center of the circle. 15. Draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. 16. Similar triangles are to one another in the duplicate ratio of their homologous sides. 17. Draw a straight line perpendicular to a plane, from a given point above it. 18. ABC is a triangle having each of the angles at B, C double of that at A, CD is drawn bisecting the angle Cand meeting AB in D; shew that BC is a tangent to the circle described about ADC. 19. Draw a circle so as to touch two given straight lines and have a given radius. 20. Two equal circles are described having the center of each in the circumference of the other, and cutting each other in the points A, B; AC is always drawn cutting the cir cumferences in D and C; shew that the triangle DBC is equilateral. VI.-ALGEBRA. 1. Find the numerical value of the expression. where a, b, c are connected by the equation a (b − c)2 − c (b + c)2 = 0. 2. Shew that the difference between the cube of a number consisting of two digits, and the cube of the number formed by changing the places of the digits, is divisible by 27. 3. Define the Least Common Multiple. State and prove the rule for finding the L. C. M. of two algebraical expressions. 4. Simplify {14 - 8 √(− 2)}. How can it be determined by inspecting such an expression whether the extraction of the outer square root will simplify it or not? 5. Define joint variation. Supposing the efficiency of a steam-engine to vary directly as the weight it can raise in a given time through a given distance, and inversely as the quantity of coal consumed in doing so; and supposing the efficiency of an engine which can raise 1 cwt. by a consumption of 1 lb. of coals to be taken as 1; what will be the efficiency of an engine, which can raise 1 ton to the same height in the same time by a consumption of 2 lbs. of coals? 6. Assuming the Binomial Theorem for positive indices, whether integral or fractional, prove it for negative. 7. Solve the equations: (1) √√(2x + 7) + √(3x + 18) = √(7x + 37). y3 5 3 (3) 4x2 + 2xy + + (4x + y) = 31, 4x − y − 14 = 0. 4 12 8. Two trains leave London at 24 hours' interval, and travel at the rate of 19 and 38 miles per hour respectively. On a certain day the first train being detained 50 minutes on the way reached Cambridge only 10 minutes before the second train was due. Find the distance of Cambridge from London. 9. A set out from Cambridge for London, walking, at the same time that B set out from London for Cambridge, on horseback. A arrived in London, and B in Cambridge, 9 and 4 hours respectively after they met. Find how many hours each required for the journey. 10. In the same time B and C together do thrice as much work as A, and A and C together thrice as much as B. Shew that C by himself can complete a piece of work in a whole number of days only when all three working together can do it in a number of days, which is a multiple of 5. TRIGONOMETRY. 1. Trace the change in the sign and magnitude of sine A as A increases from 0° to 360°. 2. Define the sine and cosine of an angle, and from the definitions, express_sin (A - B) in terms of the sines and cosines of A and B, when each of the angles A and B is greater than a right angle. 3. In every quadrilateral figure ABCD, tan A+ tan B + tan C+ tan D cot A+ cot B + cot C+ cot D = tan A. tan B. tan C. tan D. 4. If Q+= 240°, and vers = 4 vers ; find the values of and . 5. Express 666 grades in Circular Measure. 6. If the area of the polygon inscribed in a circle be th that of the polygon of the same number of sides circumscribed about it; find the number of sides in the polygons. 7. Three objects A, B, and C forming a triangle are visible from a station D at which the sides subtend equal angles. Find AD. has n values which are all different, and only n values. VII. CONIC SECTIONS. 1. Define the tangent at a given point in a curve. Prove that the tangent at any point of an ellipse makes equal angles with the focal distances. |