6. A ladder erected against an inner wall of a shed just reaches a window 18 feet from the ground, and, on being turned over through a right angle (the foot not being moved), it reaches a point on the opposite wall 7 feet 6 inches from the ground. Find the distance between the walls. 7. _D_ is the middle point of the hypotenuse _BC_ of a right-angled triangle _BAC_. Show that _DA_ = _DB_. 8. _AB_, _CD_ are two equal chords in a circle. Show that they are equidistant from the centre. 9. Show that, if two tangents are drawn to a circle from an external point, the tangents (1) are equal in length, (2) subtend equal angles at the centre of the circle. III. LATIN PROSE COMPOSITION. Translate into Latin:— When the bridge was nearly all cut away, Horatius made his two companions leave him, and pass over into the city. Then he stood alone on the bridge, and defied all the army of the Etruscans: and they showered their javelins upon him, and he caught them on his shield and stood yet unhurt. But just as they were rushing at him, to drive him from his post by main force, the last beams of the bridge gave way, and it all fell with a mighty crash into the river. While the Etruscans wondered, and stopped in their course, Horatius turned and prayed to the god of the river: ‘O father Tiber, I pray thee to receive these arms and me who bear them, and to let thy waters befriend and save me.’ Then he leaped into the river, and though the darts fell all around him, yet they did not wound him, and he swam across to the city safe and sound. IV. GREEK _AND_ LATIN GRAMMAR. (_a_) LATIN GRAMMAR. 1. Give (_a_) the meaning, gender, and genitive singular of—pes, domus, lex, opus; and (_b_) the meaning, genitive case, and comparative of—parvus, niger, vetus, senex. 2. Give the meaning, present infinitive, and 1st person sing. of the perfect indicative of—pono, capio, morior, iuvo, audeo, reor. 3. Explain what impersonal verbs are, and give instances to show the constructions used with them. 4. Translate into Latin:— (_a_) Having lost his horse, he was obliged to go on foot. (_b_) I hope that he will come quickly. (_c_) He will return home to-morrow. (_d_) I am afraid that the girl will die. (_e_) Whatever happens, we must go away. 5 What is the meaning of the following prepositions and what cases do they take—penes, coram, apud, instar? 6. What classes of verbs cannot ordinarily be used in the passive voice, and why? (_b_) GREEK GRAMMAR. 1. Give (_a_) the meaning, gender, and genitive singular of—πόλις, χρώς, κέρας, σῶμα: and (_b_) the meaning, genitive singular, and comparative of—ταχύς, μέγας, φίλος. 2. Decline in the singular—οὐδείς, ὅστις, and in the plural—σύ, οὗτος, πᾶς. 3. Give the 1st person singular of the future and of the 2nd aorist indicative active of—αἱρέω, φέρω, πάσχω, δίδωμι, θνήσκω. 4. Translate into English:—μὴ γένοιτο, πέμπτος αὐτός, οἴκαδε, οἱ σὺν αὐτῷ, ἔλεξε τοιάδε.