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9. If A, B, C be any three circles, and X, Y, Z three others connected with them by the relations that X is coaxal with B and C, Y with C and A, Z with A and B, prove that
a. The three circles X, Y, Z have the same radical centre and the same orthogonal circle with the three A, B, C.
6. When they pass through a common point they pass also through a second, the inverse of the first with respect to the circle orthogonal to A, B, C. C. When their centres are collinear, they are themselves coaxal.
10. If A, B, C be any three circles, X and X' the two circles of inversion, external and internal, of B and C, Y and Y' the two of C and A, Z and Z' the two of A and B, prove that
a. The four groups of three circles, XY'Z', YZ'X', ZX'Y', and XYZ are coaxal.
6. The limiting points, real or imaginary, of each group are the intersections of its line of centres with the circle orthogonal to A, B, C.
c. The common points, imaginary or real, of each group are inverse at once to their line of centres and to the circle orthogonal to A, B, C.
11. Prove the following reciprocal properties of homographic divi
a. A variable line intersecting two fixed lines homographically intersects every two positions of itself equihomographically.
d. A variable point subtending two fixed points homographically subtends every two positions of itself equihomographically.
b. The variable intercepted segment in the former case subtends two constant angles at two fixed points (always real). b'. The variable subtended
angle in the latter case intercepts two constant segments on two fixed lines (sometimes imaginary).
12. Prove the following reciprocal cases of involution :
a. The three sides of every triangle and every three concurrent lines through its three vertices intersect every axis in six points in involution.
a'. The three vertices of every triangle and every three collinear points on its three sides subtend every vertex in six rays in involution.
b. The six perpendiculars on the six lines from any point in the former case determine at the point a pencil of six rays in involution.
b'. The six perpendiculars from the six points upon any line in the latter case determine on the line a system of six points in involution.
HOMER AND EURIPIDES.
Translate the following passages into English :I. Beginning, Πάτροκλος δ' έπετο, σφεδανόν Δαναοϊσι κελεύων, κ. τ.λ. Ending, ώς ίπποι Τρωαι μεγάλα στενάχοντό θέουσαι.
HOMER, Iliad, lib. xvi. 372–393.
2. Beginning, ηνπερ γαρ πόλεμόν γε φύγη πολύδακρυν 'Αχαιών, κ.τ.λ. Ending, 'Αστυάναξ, δν Τρώες επίκλησιν καλέoυσιν·
Ibid., lib. Χxii. 487-5ο6. 3. Beginning, ώ θύγατερ, ουκ οίδ' εις ό,τι βλέψω κακών, κ.τ.λ. Ending, οίδεν το γ' αισχρόν, κανόνι του καλού μαθών.
EURIPIDES, Hccuba, 583-600. 4. Beginning, ου νύν με πρώτον, αλλά πολλάκις, Κρέον, κ. τ.λ. Ending, και νύν το μέν σον ού φθονώ καλώς έχειν"
EURIPIDES, Medea, 294-313. 5. Beginning, ο μέγας όλβος, άτ' αρετά, κ. τ.λ. Ending, τρώων παθέων αμοιβών.
EURIPIDES, Orestes, 797-833.
XENOPHON AND DEMOSTHENES.
Translate the following passages into English prose :-
XENOPHON, Exped. Cyri, lib. vi. c. 3. 2. Beginning, Και πρώτον μεν αμελήσαντες τού εις εκκαίδεκα, κ. τ.λ. Ending, το υπερέχον επικάμψαντες εις κύκλωσιν.
Ibid., Hist. Gracæ, lib. iv. c. 2. 3. Beginning, “Και σε γε διδάξων,” έφη, « ώρμημαι, ότι, κ. τ.λ. Ending, καταπεφρονηκόσιν οκνείς λέγειν, δεδιώς μη καταγελασθής;”
Ibid., Memorabilia, lib. iii. c. 7. 4. Beginning, Και δη και νύν τη Διοπείθει στράτευμ' έχoντι, κ. τ.λ. Ending, μικρόν πινάκιον ταύτα πάντα κωλύσαι δύναιτ' άν.
DEMOSTHENES, De Chersoneso, p. 96. 5. Beginning, 'Ανάγκη δήπου τους λόγους τούτους Αισχίνην, κ. τ.λ. Ending, ών ουδείς ουδέν ήδίκηται ιδία δήπου.
Ibid., De Falsa Legatione, p. 373.
Translate the following passages, writing brief notes on the words in italics :1. Beginning, Jam pauca aratro jugera regiæ...... Ending, Porticus excipiebat Arcton.
HORACE, Od. ii. 15. 2. Beginning, Reges dicuntur multis urgere culullis ...... Ending, Et male tornatos incudi reddere versus.
HORACE, Ars Poet., 434–441. 3. Beginning, Men. Cur non, Mopse, boni quoniam convenimus ambo, ... Ending, Sylvestris raris sparsit labrusca racemis.
VIRGIL, Ec. 4. Beginning, Chr. Revertor, postquam, quæ opus fuere ad nuptias.... Ending, Ut fert natura, facias, an de industria?
TERENCE, Andria, act iv. sc. 5.
Translate and explain the following Horatian phrases :
Mæonii carminis alite.-Cadum Marsi memorem duelli. Agentia verba Lycamben.—Marsya te voltum ferre negat Noviorum posse minoris.Levis Agyiea.—Dicitur Platus ad exemplar Siculi properare Epicharmi.Pontioa pinus. Othone contempto.-Ad imum Threx erit.—Thyna merce beatum.-Cum Velabro omne macellum.
Translate the following Terentian phrases into colloquial English :
Mansum oportuit.—Me indicente hæc non fiunt.—Utinam tibi commitigari videam sandalio caput.-Hera in crimen veniet.—Si libitum fuerit, causam ceperit, quo jure, quaque injuria, præcipitem me in pistrinum dabit. --Cyathos sorbillans paulatim hunc producam diem. Habet hæc ei quod, dum vivat, usque ad aurem obganniat. - Dare in sumptum.Omissiores ab re ne sint, metuas.- Obsonare cum fide.
Translate the following passages into English : 1. Beginning, Quemadmodum bellum minore, quam timuerant, ...... Ending, sine certaminę ingenti redit.
Livy, lib. iv. C. 43. 2. Beginning, Ob ea infensus consulibus senatus, . Ending, minus in prætura tendente.
Ibid., lib. viii. c. 15. 3. Beginning, Aut hoc judicium reprehendas tu,. Ending, pluribus verbis docere non debeo.
CICERO, Pro Clurutio, c. 41. 4. Beginning, Venio nunc ad Dorylensium testimonium :. Ending, nobilissimum civem vindicetis.
Ibid., Pro Flacco, c. 17.
Translate the following passage into Greek Prose :Beginning, I am very much pleased with a consolatory letter of Phalaris, .. Ending, You must first see us die, said he, before that question can be answered.
ADDISON, Spectator, No. 349. Translate the following passage into Greek Trimeter Iambics :Beginning, Iphigenia.
Conclude the tale.... Ending, And lives Electra, too?
From GOETHE, Iphigenia in Tauris, act ii. sc. I.
Translate the following passage into Latin Verse :Beginning, The more we live, more brief appear.... Ending, Proportion’d to their sweetness.
CAMPBELL, The River of Life.
Translate the following passage into Latin Prose : Beginning, No hero of ancient or modern days can surpass... Ending, to witness that he dies without a groan.
IRVING, The Sketch-book- Traits of Indian Character.
LATIN HISTORY, GEOGRAPHY, AND GRAMMAR.
1. Who were the Senatores Pedarii and the Quæstores Classici ? 2. What were the duties of the censors, and what checks were imposed
3. Show that the constitution of Servius was formed on an Hellenic model.
4. Sketch the campaigns of Cæsar in Greece and Spain. 5. Give a short account of the Perusian War.
6. Who were the contending parties in the battles of Asculum, Aquæ Sextiæ, and Thapsus ?
7. With what measure did Camillus begin his censorship?
8. What fact in Roman history is proved by the origin of the words, calx, machina, groma, and clathri?
9. What is the origin of the word vates? Give some other words in the Latin language drawn from the same source.
10. Explain the formation of the words, ussi, gessi, sero (sevi), vidi, pegi, Tarentum, statera, and placenta.
11. How are questions expressed in the oratio obliqua ?
14. Draw a map of Sicily, marking the chief towns along the coast, and giving their modern names.
15. Give the ancient and modern names of the chief rivers of Spain.
1. Mention the chief ties which bound the Greeks together.
4. Mention briefly the principal events from the battle of Eurymedon to the thirty years' truce.
5. Mention the events from the battle of Cnidus to the peace of Antalcidas.
12. Name in order, beginning from the most western, the divisions of central Greece.
13. Where were the following towns :—Pagasæ, Byzantium, Pylus, Larissa, Eleusis, Potidæa, Eretria, Naupactus, Amyclæ?
14. Where were the rivers-Achelous, Halys, Eurotas, Mæander, Peneus, Alpheus, Hebrus?
15. Where were the mountains—Parnes, Taygetus, Othrys, Erymanthus, Pindus, Oeta ?
ENGLISH LANGUAGE AND COMPOSITION.
1. Assign derivations for any of the following words :-Climate, gossip, palfrey, bide, pedlar, pelican, prodigious, prose, reputation, steward, stratagem, tadpole.
2. To what historical events may the origin and present signification of the following words be ascribed :--Assassin, caitiff, poltroon, saunterer, dunce, bigot?
3. "In a vast number of instances a word lives on as a verb, but has ceased to be employed as a noun--or, with a reversed fortune, it lives on as a noun, but has perished as a verb-or, again, the affirmative remains, but the negative is gone--or, once more, with a curious variation from this, the negative survives, while the affirmative is gone.” Give two or three examples of each of these cases.
4. Spell the following words so as to indicate their etymologies :—Diamond, grocer, woodbine, scent, whole, afraid, country-dance, necromancy.
5. Meaning of the word trivial in the following quotation :-“ Æquitas optimo cuique notissima, is a trivial saying, A very good man cannot be ignorant of equity.” 6. Write a short essay on the lines
Strange cozenage ! None would live past years again;
What the first sprightly running could not give.”