8. Reduce to its simplest form the expression sec 210.sec 0 12. Eliminate 0 and 10. Prove the formula which gives the area of a triangle in terms of the three sides. 11. In a triangle if a3, b2, c2 be in arithmetical progression, sin 3B sin B = and hence prove that a cos A+ b cos B+ e cos C 2abc π I I ( from the equations cos 20 m, b sin 2p + a cos 2p = n, 13. The length of an arc of 45° in one circle is equal to that of 60° in another; find the circular measure of an angle which would be subtended at the centre of the first by an arc equal to the radius of the second. 14. If 1-cos 2A - cos 2B - cos 2C-2 cos A cos B cos C=o, prove that A+B+C = (2n + 1) π. MR. LESLIE. 15. Prove that sin A sin B sin (A – B) + sin B sin C'sin (B - C) + sin C'sin A sin (C– A) = } { sin 2 (A – B) + sin 2 (B - C) + sin 2 (C-A)}. a2 c212 16. Calculate loge 2, and loge 5 to three places of decimals; and hence show that if M be the modulus of the ordinary system of logarithms, m 2ac 17. Prove geometrically that if m / 1, I tan-1 tan-i = 4 tan-1 = 2.3 nearly. I 5 π 4 18. Find the values of x which satisfy the equation I 17/2 I I + m tan-1 sin 3x + cos 3x= I 239 D. DR. SALMON. 1. Every circle which cuts two diagonals of a quadrilateral harmonically, cuts the third harmonically? 2. In any triangle, the three chords of intersection of the circumscribing circle, with the exscribed circles, determine a triangle in perspective with the original? 3. Given base and sum of sides of a triangle, find the locus of the intersection of either side with the perpendicular from middle point of base on the external bisector of the vertical angle. 4. Given base and sum of sides, find the locus of the intersection of the bisector of base, with the perpendicular from either extremity of base on the external bisector of vertical angle. MR. STUBBS. 5. All the circles having their centres on the same right line, and cutting at right angles a given circle, have the same radical axis; and these circles taken two and two with the given circle have the same radical centre? 6. If c, c', c", c" be the centres of the inscribed and three exscribed circles of a triangle, the middle points of the six lines joining these centres lie on the same circle? 7. If A, B, C be three points on a line disposed in any manner, and AP'; BP', CP' the three lines connecting them with any point P', AP'2 BP'2 CP'2 + prove this, and deduce from it by producing all the lines P'A, P'B, P'C to A'B'C' until the rectangles P'Ax P'A' = P'B. P'B' = P'C. P'C' = k2 that if four points P', A, B, C' lie on the circumference of a circle, P'B'.P'O' P'A'. P'C' P'A'. P'B' 8. A line BOC is drawn through a fixed point O cutting the legs of an angle BAC in B and C; prove that the area of the parallelogram formed by drawing parallels through O to the given lines AB, AC is an harmonic mean between the areas of the triangles, BOA, COA. = 1; = 1. MR. LESLIE. 9. If from the intersection of the diagonals of a quadrilateral inscribed in a circle perpendiculars be dropped on the sides, the quadrilateral formed by joining their feet is, of all quadrilaterals inscribed in the given one, one of minimum perimeter? 10. If from a point on one circle a tangent be drawn to another, prove that the square of the tangent is in a constant ratio to the rectangle under the chords joining the point to the intersections of the circles. 11. If from any point of a right line perpendiculars be dropped on the sides of a triangle, prove that the locus of the intersection of the bisectors of the sides of the triangle formed by joining their feet is a right line. 12. Prove that the area of the triangle formed by the polars with respect to the inscribed circle of the middle points of the sides of a triangle is equal in area to the original. Classics. ESCHINES AND DEMOSTHENES. MR. GRAY. Translate the following passages into English : - 1. Beginning, μήθ ̓ ὑμεῖς ποτε εἰς τοσαύτην ἀταξίαν, κ. τ. λ. Ending, καὶ μοι λέγε τοὺς νόμους. ESCHINES, Adv. Ctesiph., 38. 2. Beginning, ἐπειδὴ δὲ σιγᾷς, ὅτι μὲν ἀπορεῖς, κ. τ. λ. Ibid., 166. 3. Beginning, ἐγὼ λοιδορίαν κατηγορίας τούτῳ, κ. τ. λ. Ending, ὅρα μὴ τούτων μὲν ἐχθρὸς ᾖς, ἐμὸς δὲ προσποιῇ. DEMOSTHENES, De Coronâ, 123–125. 4. Beginning, Ορᾶτε δὲ κἀκεῖνο, ὦ ἄνδρες ̓Αθηναῖοι, κ. τ. λ. Ending, μετ ̓ ἰσηγορίας ζῶν χρημένοις εὖνοι γένοιντ ̓ ἄν. Ibid., p. 194. 5. Beginning, ̓Αμφότεροί μοι δοκοῦσιν ἁμαρτάνειν, κ. τ. λ. Ending, διαβεβλήσεται δὲ πρὸς ἀμφοτέρους· Ibid., p. 202. 1. A signal act of violence by Sparta is marked as the turning point of her fortunes? 2. It was Sparta, too, that prepared the indispensable basis upon which Philip afterwards rose ? 3. Describe the three great battles of Alexander, and point out the respective results of these victories. 4. Give an account of the Lamian war. 5. Give the special rules for the use of ov and μý. 6. What is the difference in meaning between exεiv and oxεiv? and give the general principle on which this difference depends. 7. As indicating time, how does the indicative mood differ from the other moods? 8. Under what conditions only can attraction of the relative pronoun take place? 9. State exactly the ways in which compound verbs are formed. 10. Mention the principal towns in the island of Euboea, and give a rough map showing their relative positions. 11. Why was it that the seizure of Elateia caused such consternation at Athens ? 12. Explain fully καταβαλόντα ἐᾷν ἐν ὑπωμοσίᾳ, and προθεσμία. CICERO'S ORATIONS. MR. POOLE. Translate the following passages into English Prose :1. Beginning, Quod ubi auditum est, æstuare illi,. . . . Ending, ait se velle de illis H-S LXXX. cognoscere. In Verrem, act II. lib. ii. c. 23. 2. Beginning, Tot prætores in Sicilia fuerunt; Ending, sed postulationum formulas usitatas. 3. Beginning, Opus in acceptum retulit quadriennio post,. Ending, "Recita nomina Rabonii." Ibid., c. 60. Ibid., act II. lib. i. c. 57. 4. Beginning, Gracchorum potentiam majorem fuisse arbitramini,. . . . Ending, illi parati sint ad nutum futuri? Philip. vii. c. 6. I.-a. Notice the use of "tamen," and the construction of the clause "ut sibi. . . curet" in the first passage. State the different constructions used with "cavere," and their meanings. b. Explain the clause "ne in cellam quod imperatur æstiment" in the second passage; noticing the various forms under which the Romans obtained corn from the Provinces. c. Translate the following passages accurately : Mancipes a civitatibus pro frumento pecuniam exegerunt. Quid hoc Verre prætore factum est solum? Non; sed etiam quæstore Cæcilio. Quid igitur daturus es huic crimini quod et potuisti prohibere ne fieret et debuisti? an totum id relinques. Iratus iste vehementer Sthenio et incensus hospitium renuntiat; domo ejus emigrat atque adeo exit: nam jam ante migrarat. Herodotus quum Roma revertitur diebus, ut ipse putabat, quindecim ante comitia, offendit eum mensem qui consequitur mensem comitialem, comitiis jam abhinc diebus xxx factis. d. Explain the word "mancipes;" and give the syntactical construction of the clause "quid igitur ... debuisti." Distinguish the words "exit” and “emigrat;" and give the force of "atque adeo." Write a note explanatory of the meaning of the last passage. 2. What campaigns not long before the first Punic war seem to have established the Roman supremacy in Italy? Give their dates, and state what remarkable men were engaged on both sides? 3. What was the provision of the Lez Plautia Papiria? and show how it tended to promote the interests of Rome. 4. Merivale remarks that Julius Cæsar showed his design of forming the Roman dependencies into one united whole by several of his measures? 5. Point out some of the marks of the later despotic empire appearing in the case of Julius Cæsar. 6. Give a short account of the circumstances which immediately preceded and followed the treaty of Brundusium, made in A. U. C. 714. 7. To what circumstances does Sallust attribute the general corruption of men's minds which prepared the way for Catiline's conspiracy? 8. Point out the advantages arising from the "edictum ;" and explain the terms "perpetuum” and “tralaticium,” which were used in reference to it? 9. Give some account of the government of Sicily, noticing how justice was administered, and how the various charges were defrayed? 10. What were the various classes of rights enjoyed by a Roman citizen possessing the full franchise; and explain the terms “peculium,” “dominium,” and “possessio,” as regarded property ? 11. Draw a map of Italy, marking on it some of the most important towns which were besieged during the Social war. 12. a. How is the tense of the verb in the conjunctive mood determined in a dependent clause? b. Distinguish the senses of the conjunctions "sed," "autem," "immo;" and notice the difference in the use of "et” and “que." MR. LONGFIELD. Translate the following passage into Latin Prose :— The history of the war against the Veientes, Capenates, and Falisci, and of the siege of Veii, which is said, like that of Troy, to have lasted ten years, rests on evidence little reliable. Legend and poetry have taken possession of these events as their own, and with reason; for the struggle in this case was waged, with unprecedented exertions, for an unprecedented prize. It was the first occasion on which a Roman army remained in the field summer and winter, year after year, till its object was attained. It was the first occasion on which the community paid the levy from the resources of the state. But it was also the first occasion on which the Romans attempted to subdue a nation of alien stock, and carried their arms beyond the ancient boundaries of the Latin land. The struggle was vehement, but the issue was scarcely doubtful. The Romans were supported by the Latins and Hernici, to whom the overthrow of their dreaded |