Изображения страниц



1. If the weight of a cubic inch of water be 252.5 grains, calculate the radius of a sphere of gold which will weigh 10 lbs. avoirdupois, the weights of equal bulks of gold and water being as the numbers 19.35 : I.

2. What is the cost of lining a cistern with lead, 4th of an inch thick, if the dimensions be 6 ft. long, 2 ft. wide, and 3 ft. high, at 4d. per lb., the weights of equal bulks of lead and water being as 11.35:1 ?

3. A, B, and C enter into partnership: A contributes £248 for 10 months, B £670 for 3 months, C £512 for 6 months. Required the share of each in a gain of £439 188. 8d.

4. The rate of exchange between Berlin and London is 6 thalers 25 groschen for £1, and between London and Paris 24 francs 75 centimes. What is the rate of exchange between Paris and Berlin? [30 groschen = 1 thaler.]


5. Four points A, B, C, D on a line are distant from a fifth point 0 on the line by intervals = 5, 6, 11, 17 ft. respectively; required the distance from O of the point P for which PA. PB = PC. PD.

6. In the same case, required the distance from O of the point P for which PA2+ PB2 + PC2+ PD2 is the minimum, and with it the value of the minimum sum.

7. The three sides of a triangle are 13, 17, 21; calculate, to three decimal places, the radius of its inscribed circle.

8. In the same case, calculate, to one place of decimals, the distance between the centres of its circumscribed and inscribed circles.


9. A person buys 100 shares in a company for £3500; after receiving four half-yearly dividends of 158. 4d., 208. 10d., 308. 4d., and 388. 9d. per share, he sells at a profit of 43 per cent. Reckoning the simple interest of money at 4 per cent., how much above that interest has he gained?

10. When the 3 per cents. are at 92, and the 3 per cents. at 81, and a stock paying 4 per cent. is at 105, which would be the best investment, and which the worst? And if the sum invested is £13,125, what is the annual income in the most favourable case?

II. A man invests £1000 equally in the shares of two railway companies; the shares of one are at 3, and the other at 5 per cent. discount; the price of stock in the former suddenly rises 7 per cent., and that of the latter is depressed 5 per cent. If he now sell out, what will he gain or lose?

12. Calculate, to one place of decimals, the value of g from the following equation:

[ocr errors]



when R20888700, r = R× 59.9643, P=27.3217, π= 3.14159.



1. If two tangents be drawn to a circle, the perpendicular from any point of the intercepted arc upon the chord of contact is proportional to the chord of the circle which passes through that point and the intersection of the tangents?

2. Given of a triangle the base and vertical angle; find the locus of a point on the bisector of that angle whose distance from the vertex is half the sum of the sides.

3. Given of a triangle the bisector of the vertical angle, and the rectangles under each side and the conterminous segment of the base made by that bisector; construct the triangle.

4. If the diameter MN of the circle circumscribing a given triangle ABC be drawn perpendicular to the base at 0, and a perpendicular BP be dropped upon this line from the vertex, then will the square of half the difference of the sides AB, BC, be equal to the rectangle under the segments MO, PN, of the diameter between the circle and the base, and between the circle and the foot of the perpendicular BP.


5. If two points be mean centres of the same system of points for the same system of multiples, all points indifferently possess the same property?

6. Determine the point from which if perpendiculars be let fall upon any number of given lines, the sum of their squares shall be the mini


7. Construct a quadrilateral of given species whose vertices shall be on given lines; and explain the circumstances under which the problem becomes indeterminate.

8. Two lines inclined at an angle of constant magnitude revolve round a fixed point; required the envelope of the circle circumscribing the triangle they determine with a fixed line.


9. If all the sides but one of a polygon inscribed in one of a coaxal system of circles touch given circles of the system, that one will also touch a circle of the system?

10. State the condition necessary that the perimeter of a polygon whose vertices lie on given lines may be a minimum; and show that it is not possible to construct a quadrilateral satisfying this condition, unless the given lines form a quadrilateral inscribable in a circle.

II. Prove that the circle which passes through the feet of the perpendiculars of a triangle touches the inscribed and exscribed circles of the triangle.

12. Given the difference of the segments of base made by the perpendicular, and difference of base angles; construct the triangle so that the sum of the squares of the three sides may be a maximum.

[merged small][ocr errors][merged small]


Translate all the following passages into English Prose :-1. Beginning, ἤδη δ ̓ ἑσπέρας οὔσης καὶ σκότους, κ. τ. λ. Ending, καὶ ἀνώμοτος ἀπηνέχθη.

Meidias, 542.

2. Beginning, ἐμοὶ μέν, νὴ τὸν Δία καὶ τὸν ̓Απόλλω καὶ τὴν, κ. τ. λ. Ending, τόν γε δὴ μέχρι τῆς κρίσεως χρόνον, εἰ καὶ μὴ πάντα ; Ibid., 578. 3. Beginning, ̓́Ετι τοίνυν, εἰ μὲν μηδένα μηδὲν ἔχοντα εἴχομεν, κ. τ. λ. Ending, πρὸς δ ̓ ἐκεῖνον ἐλθὼν κρίνεσθαι βούλεσθαι; De Falsa Leg., 375

4. Beginning, ἔτι τοίνυν, ὅταν μὲν ἴδητε δεινότητα, κ. τ. λ. Ending, δωροδόκου καὶ οὐδ' ὁτιοῦν ἐροῦντος ἀληθές.

Ibid., 450.

5. Beginning, "Ισως τοίνυν καὶ τοιαύτην τινὰ ἐροῦσιν ὑπόνοιαν, κ. τ. λ. Ending, ὅτι τῷ φενακίζειν οὐκέθ ̓, ἃ βούλεται, πράξει. Adv. Aristocrat., 685.

6. Beginning, τί δ ̓, εἴ τις δίκην ἐξούλης αὐτῷ λαχὼν μηδὲν, κ. τ. λ. Ending, κἂν μάρτυρας ὑμῖν παρεσχόμην. Adv. Baotum, 998.

I. a. Write notes upon the meaning of the terms, ἀποδεδιητημένην, ἀντιλαχών, and ἀπηνέχθη; and explain the force of οἷός τ ̓ ἦν in the first passage ; and of ἀπήλλαγμαι in the second.

6. Explain the force of the words ξυληγῶν and καταλλαττόμενος, in reference to the charge brought, in the third passage; and the different senses of the preposition ἐπί, and the meaning of the words παντὸς ἥττονος λήμματος, in the fourth.

2. State some of the methods adopted by the Athenians to prevent vexatious litigation.

3. What were the classes who composed the state of Sparta? Describe accurately their social positions; and explain the terms Ομοιοι, Ὑπομείονες, and Κρυπτεία?

4. What were the circumstances under which Cleisthenes succeeded in establishing freedom at Athens; and the dangers which immediately afterwards threatened to subvert it?

5. How was Philip first brought into collision with the Phocians? State how he finally succeeded in overcoming them, and the Olympiad in which he did so?

6. Write an account of the life of Antipater: stating also the first distribution of the empire of Alexander.

7. What was the nature of the Amphictyonic Council, its design, and the origin and date of the first Sacred War?

8. Draw a map of the Persian Empire, marking on it the positions and names of the leading cities captured by Alexander.

9. In what verbs is the use of the future middle in an active sense most common? Give examples of common verbs which have no tense characteristic in their future.

10. How are adverbs formed from verbal,and substantival stems, and how are the different forms accented?

11. What are the peculiarities of the infinitive, when used as an imperative in Greek? What are the tenses of the Latin verb to which the aorist indicative corresponds? and why?

12. When is où admissible in dependent sentences? and state in what cases un is used with the optative and indicative moods.



Translate the following passages:

1. Beginning, Factus es ei rursus, nescio quomodo, familiaris..... Ending, sed priusquam habeantur, debet nuntiare.

Phil. ii. c. 32.

2. Beginning, Quamquam mihi semper frequens conspectus vester. Ending, rei fructum suo judicio tribuendum esse duxerunt. De Lege Manilia, c. 1.

3. Beginning, Sed, ut omittam communem caussam, . Ending, obtinere voluisse, quam aliquem se maluisse.

Pro Q. Ligario, c. 7.

4. Beginning, Et primum M. Catoni, vitam ad certam rationis..... Ending, cui video maximas reipublicæ tempestates esse subeundas? Pro L. Murena, c. 2.

5. Beginning, Hic ait se ille, judices, regnum meum ferre non posse... Ending, illud quæro, peregrinum cur me esse dixeris. Pro P. Sulla, c. 7.

1. Under what law was Archias prosecuted, and what was his defence? 2. Give an account of the origin, extent, and constitution of the Roman province of Asia; mentioning its principal towns, rivers, and products. What place formed the great entrepôt for the commerce between Europe and Asia at this time?

3. Explain operæ subseciva, legibus solutus, congiarium.

4. What is the exact meaning of such adjectives as Hispaniensis?

5. What are the principal classes of verbs derived from verbs?-how are they severally formed?-and give examples of each. When only is a verb found in the first part of a compound word?

6. What is the rule for the construction of the names of towns in answer to the question where ?-and how may this rule be accounted for?

7. When do verbs signifying "to remember or forget" govern the accusative?

8. When is the ablative used after comparatives?

9. How in Latin do you express a wish for a thing regarded as no longer attainable?

10. When must ut nemo, &c., be used instead of ne quis, &c. ?—and when is the gerundive used instead of the gerund?

11. Explain the Roman method of counting money, and account for such phrases as syngrapha sestertii centies.

12. Mention the principal actions of Appius Claudius Cæcus.

13. State the principles on which the Italian nations were so settled as to remain the peaceable subjects of Rome.

14. The division of the Italian communities, as established by the Roman Government, was threefold? State exactly the nature of each.

15. How were the Romans led to form the province of Transalpine Gaul?

16. Give an account of the career of the Teutones and Cimbri till their overthrow by Marius.

17. Give an account of the legislation of Sulla, and state of Rome and the provinces at the close of the first civil war.


Translate the following passage into Latin Prose :


It was not till the Gauls had crossed the Tiber, and were at the rivulet of the Allia, less than twelve miles from the gates, that a Roman military force sought to hinder their passage. And even now they went into battle with arrogance and fool-hardiness,—not as against an army, but as against freebooters under inexperienced leaders, Camillus having, in consequence of the dissensions of the orders, withdrawn from taking part in the affairs. Those against whom they were to fight were but barbarians; what need was there of a camp or of securing a retreat? These barbarians, however, were men whose courage despised death, and their mode of fighting was to the Italians as novel as it was terrible; drawing their swords, the Celts precipitated themselves with furious onset on the Roman phalanx, and shattered it at the first shock. Not only was the overthrow complete, but the disorderly flight of the Romans, who hastened to place the river between them and the pursuing barbarians, carried the greater portion of the defeated army to the right bank of the Tiber and towards Veii. The capital was thus left to the mercy of the invaders; the small force that was left behind, or that had fled thither, was not sufficient to garrison the walls, and three days after the battle the victors marched through the open gates into Rome.-MOмMSEN, History of Rome.

« ПредыдущаяПродолжить »