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15. Describe a circle which shall pass through a given point, and bisect the circumferences of two given circles.
16. Define ratio, and show how any number of ratios can be compounded into a single ratio.
17. Find a point in a given line, such that the sum of its distances from two given points shall be equal to a given line.
18. State Euclid's twelfth axiom, Book I., and the use which he ma of it Show that the following would equally have suited his purpose, viz. :-Through a given point there cannot be drawn more than one right line, which shall be parallel to a given line.
1. The three sides of a triangle are 23.4, 25.2, and 27; calculate the radius of its circumscribing circle.
2. The three perpendiculars of a triangle are 18.2, 19.5, and 21; calculate its area.
3. The radii of two circles are 84.5 and 97.5, and the distance between their centres is 91; calculate the length of their common chord.
4. The radii of three circles which touch two and two are 13.5, 15.75, and 18; calculate the radius of their common orthogonal circle.
5. A merchant commenced trade with £3000; he settles his accounts half-yearly, and finds he clears 3 per centeach half year on the entire stock’ in trade. What is he worth at the end of 33 years ?
6. A fast train leaves Bristol for London (120 miles distant) at 2 P. M., and travels at 25 miles an hour. At what time must a luggage train, which vels at the rate of 15 miles in 50 minutes, have left, so as to escape being overtaken by the fast train ?
7. What ready money will discharge a debt of £528 98. due 4 months hence, at 43 per cent. ?
8. A and B can do a piece of work separately in 12 and 16 days, respectively; they work together at it for 3 days, when A leaves it, but B continues, and after 2 days is joined by C, and they finish it together in 3 days. In what time would do it alone ?
9. Find accurately to four decimal places the value of the fraction
0.0'31 10. One hundred weight of flour yields 36 quartern loaves: calculate the price of the 2 lb. loaf, the price of flour being 528. 6d. per sack of 2 cwt., 58. per sack being the expense of manufacture, and io per cont. being allowed for profit.
11. The sides of a triangle are 4, 5, 6: calculate to three places of decimals the distance between the centres of its inscribed and circumscribing circles.
12. The bisectors of the sides of a triangle are 2, 3, 4; calculate the sides to four decimal places.
1. Draw the shortest chord of a given angle that can be cut in a given ratio by a given line.
2. Find the point on a given line from which, if perpendiculars be let fall on three other given lines, the area of the triangle determined by their feet shall be the minimum.
3. In a given circle inscribe a triangle two of whose sides shall touch a second given circle, and the third pass through a given point.
4. Given of two circles a pair of common tangents, and a common orthogonal circle; construct them.
5. Find a point in the diameter of a semicircle such that drawing a secant from it, and letting fall on the diameter perpendiculars from the two points of intersection; the ratio of these perpendiculars may be given, and also the length they intercept on the diameter.
6. Divide a triangle by a right line from vertex to base, so that the circles inscribed in the two triangles thus formed may be equal.
7. Each of the internal diagonals of a quadrilateral divides it into two triangles ; take the intersection of the bisectors of the sides of each of the four triangles thus formed. Show that the quadrilateral made by joining these four points is similar to the given one, and equal to the ninth part of its area.
8. If two chords of a circle be parallel, any circle which passes through the points in which the lines joining their extremities directly and transversely intersect, cuts the given circle orthogonally.
9. Prove that the rectangle under the diagonals of a quadrilateral can never be greater than the sum of the rectangles under the opposite sides. In what case is it equal to the sum ?
10. A and B are given points on a diameter of a circle ; through B draw a chord such that the triangle with its vertex at A, and chord for base, shall be equal to a given area. Find the limit to the magnitude of the area, and discuss the two cases which may occur.
1. The hypothenuse of a right-angled triangle is given; find the locus of the middle point of the line joining the vertices of equilateral triangles described on the sides.
12. The sides of a triangle of given species pass through three fixed points; find the locus of the intersection of its perpendiculars.
κ. τ. λ.
Translate the following passages into English Prose :I. Beginning, ότι μεν δή λιποταξίου γραφήν κατεσκεύασε κατ' εμού, Ending, ουδεμιάς έγωγ' έτι, προσδέομαι δίκης, αλλ' έκανήν έχω.
Contra Meidiam, 547, 548. 2. Beginning, Τούτο μέντοι το τοιούτον έθος και το κατασκεύασμα, Ending, οι βδελυροί και χρήματ’ έχοντές [είσιν εγγυτάτω].
Contra Meidiam, 555. 3. Beginning, Φέρε δή και όσους αυτός έλυσάμην τών αιχμαλώτων, Ending, λέγε ταύτας τάς μαρτυρίας.
De Falsa Legatione, 394. 4. Beginning, Καίτοι σκέψασθε όπως επί των προγόνων ταύτ' είχεν:
κ. τ. λ. Ending, δεδωκατε επ' αυτούς εκάστω τούτων.
De Contributione, 172.
1. How was the Macedonian Empire divided on the death of Alexander ?
2. Give an account of the Chremonidean war.
3. What was the constitution of the Ætolian league? How and when did this league obtain power ?
4. Enumerate some of the chief opinions held concerning the origin of the Ephoralty at Sparta ? Grote mentions an ancient custom pointing to the popular origin of this magistracy?
5. What were the causes of the decay of Sparta ?
6. What reforms have been generally ascribed to Lycurgus? Who originated the idea that he redistributed the land among the citizens? At what time is it most likely that he lived? What mistakes are made by Plutarch with respect to him?
7. What is the first piece of true history relating to Thebes ? 8. What are the remarkable points in the history of Greece from B. C. 776 to B. C. 560
9. What effect had the configuration of Greece upon the political relations of the inhabitants ?
10. Give some examples of the demonstrative use of the Article in Attic Greek.
11. When is the subject not marked by the Article ? 12. Give an account of the ways in which äpa, ye and åv are used.
13. What are the roots of the following words, and how do they appear in Greek :-friend, thin, ankle, awe, hemp, and grow ?
14. What were the original forms of the accusative plural of the Greek noun and of the first aorist Participle Active, and illustrate these forms from the Greek dialects and the kindred languages ?
15. Compare the case endings of the Greek with those of the Latin
16. What were the original sounds of ε and o? 17. Trace the steps by which the following pairs of words are connected with each other :-κώς, πώς; βανα, γυνή, βάω, go; and κλυτός, Ioud.
Translate the following passages into English :1. Beginning, Nam et publicani, homines et honestissimi et ornatissimi, .. Ending, neque aliis voluntas propter timorem.
Pro Lege Manilia, c. vii. 2. Beginning, At domicilium in Italia non habuit is, ... Ending, id huic, summa ingenii prædito gloria, noluisse !
Pro Archia Poeta, c. iv., v. 3. Beginning, Ecce Dolabellæ comitiorum dies : sortitio prærogativæ : Ending, quæ necesse est aliquando ad nostrum collegium deferantur.
Philippica Secunda, c. xxxiii. 4. Beginning, M. Fulcinius fuit, recuperatores, . Ending, ut frueretur una cum filio.
Pro A. Cæcina, c. iv.
1. What are the three forms of the perfect active in Latin ? Give examples from each of the four conjugations of the forms that occur in them.
2. Conjugate the verbs from which the following participles come : amictus, refertus, oblitus, fissus; also, sero to plait, sero to sow, texo, comminiscor. Conjugate and give the meaning of metior, mēto, měto; operor, operio, opperior.
3. Distinguish between frustra, nequidquam, and incassum; and between aut, vel, and -ve.
4. When is quum 'when' followed by the indicative, and when by the subjunctive? When does quum since, because,' take the indicative, and when the subjunctive ?
5. Trace the course of the Via Appia, marking the principal towns. When and by whom was it constructed ?
6. Explain the following words and phrases : prærogativa, obnuntio, prævaricator, inquilinus, sectio, congiarium, legibus solutus, ego amplius sestertium ducenties acceptum hereditatibus rettuli.
7. Quote Cicero's character of Cæsar as given in the second Philippic. He also states the importance of the island of Delos at that time?
8. Why was the prerogative of presiding in the Judicia so much coveted!
9. Trace the steps by which the Roman province of Transalpine Gaul was formed, and mention the principal colonies in it.
10. Give an account of the revolt of Saturninus. Merivale explains a certain obscurity in the accounts history has given of it ?
11. State clearly the peculiar privileges of the jus Latii and the jus Italicum.
12. Mention, with their geographical boundaries, the provinces of the Roman empire under Sulla.
13. The various relations in which the different classes of the provincial population stood to the ruling city, have been compared with the constitution of a Roman household ?
14. Sketch briefly the earlier career of Pompeius, and give an account, at some length, of his exploits in the East.
Translate the following passage into Greek Prose :-
SPECTATOR, No. 634.
Translate the following passage into Greek Verse :-
TAYLOR, Philip van Artevelde.
Translate the following passage into Latin Lyric Verse :-
Translate the following passage into Latin Prose :-
LIDDELL, History of Rome, book vil. c. xliv. $ 19,