THE POPULAR EDUCATOR. 



EXERCISE 93. ENGLISH-GREEK. 



1. They will be slain. 2. They were slain. 3. He was slain. 

 4. Two soldiers were slain. 5. Many men will be slain. 6. I 

 shall bo educated. 7. He will be educated. 8. We shall be 

 educated. 9. Ye two will be educated. 10. I was well edu- 

 cated. 11. The constitution was destroyed. 12. The constitu- 

 tion will be destroyed. 13. The treaty was broken. 14. The 

 treaty will bo broken. 15. The treaty being 1 broken, the citizens 

 were slain. 16. The robbers were slain. 17. The robbers are 

 eaid to have been slain. 18. The democracy will be destroyed. 



KEY TO EXERCISES IN LESSONS IN GEEEK. XXX. 



EXERCISE 84. GREEK-ENGLISH. 



I. The soldiers have slain two thousand two hundred and sixty-five 

 of the enemy. 2. Pherecydes used to say that he had sacrificed to no 

 god. 3. As you are (having been produced) young, learu many good 

 things. 4. The soothsayer has foretold the future well. 5. You have 

 instructed your children well. 6. Medea, having slain her children, 

 rejoiced. 7. The Lacedaemonians had destroyed Platoea. 8. Sardana- 

 palus had put on a woman's garment. 9. When the sun had set, the 

 enemy approached. 10. Alexander, in his pursuit of (pursuing) Darius, 

 the king of the Persians, had made himself master of great wealth. 



EXERCISE 85. ENGLISH-GREEK. 



1. I\f 



a. 2. \\etjtovevnaJiv. Z. V.ire<j>ovtvntt. 4 I'ovcucrowo-iv. 5. 

 V,Q>ovev(rev G. Qovevao/Afv. 7. nc^oi/ei/Ka/ie . 8. .irf<povtvneifiev. 9. Of 

 0-owom 10. Te0wea<r<i/. 11. EreffuKfirav. 12. E9u<rav. 13. 'O navris ry 0e<a 



eSv<rev. 14. *O fAavrtt ^tf> Ofy /3ouc 4(CaToc reOuxev. 15. llauWu TO reKva. 16. 

 Cn-aiAd'ov rm TCKKI. 17. Tlcuievffu ra TtKKo. 18. ETOiSei/era Ta rent/a. 19. 

 'fleiraiiei/ica ra reKva. 29. Eireraii?ei/eii' ra rexva. 21. AXefavipot Ba0v\iava 

 nare\vnfv. 22. AXefavdpot \\uftv\u>va KareXeXixcet. 23. 'O Trait <r-ro?.t\v 



yvvaiKetav ev&uet. 24. 'O iraif oroXflv fuvaiKtiao evit&vKev. 25. 'O jrait (TToXrji' 

 TwatKf-iuv ece<5eii>Ke<. 23. 'O irait aro\n fmatKetav tvbvarfi. 



EXERCISE 86. GREEK-ENGLISH. 

 1. Two men are fighting. 2. Let us fight bravely for our country. 



3. It is necessary that a son should obey his father. 4. Many good 

 men are poor. 5. It is honourable to obey the laws of the country. 



6. Do not welcome those of your friends who gratify you in bad things. 



7. Let each go quietly along the middle of the road. 8. Let the citizens 

 obey the laws. 9. My two brothers follow me. 10. If you are willing* 

 to do well, work. 11. If you wish* (should you wish) to do well, work. 

 12. No one who lies is concealed for a long time (i.e., no one lies for a long 

 tim without being found out). 13. The Lacedaemonians used to go on 

 their expeditions to the sound of flutes. 14. Would that all would 

 consult without anger. 15. Two beautiful horses were driven into the 

 city. 16. If you are poor you have few friends. 



EXERCISE 87. ENGLISH-GREEK. 



1. EKEIVO? ireverai xat cAi^ow ^1X01/5 ex c< - 2. llevofjiai. 3. E/3ou\ei6TO. 4. 

 E/3oi'Xei/(r0n'. 5. Bou\ei>o^iai. 6. BouXeveraf. 7. BovXei KaXur irparretv, 

 ep^afow. 8. Ea /?oi/\t; Ku\w irparretv epfafow. 9. EpfafocTai. 10. KaXwc 

 epyaftToi. 11. Eipya^ero. 12. Eip 7 afe<r0e. 13. EipjattaOov. 14. Ep-rfo- 

 fieOa. 15. MOXOMHI. 16. Efaaxonnv. 17. E/iaxoi-TO. 18. MOXOVTOI -yevvaiiat. 

 19. Max7#e. 20. Efjiaxeatie. 21. fl arpartiarat fexi/cuwt /j.ax,fo0t irfpi rijt 

 roTpi3( r. 22. KaXov ec-n jrepi TK varpiiot nuxeotiau. 23. 2oi firo^ai. 24. 

 'BfjLoi f jrerai. 25. E/uoi evovrai. 26. Ty a-rpurtjiy tiro/ueCa. 27. Tut urpa- 

 -TewjuaTi cin-ojue0a. 28. Toir vofiois, ta iraiAet, eireatte. 



EXERCISE 88. GREEK-ENGLISH. 



1. The robbers have been slain. 2. Two brothers have been edu- 

 cated by the same master. 3. The monarchy has been destroyed by 

 the people. 4. Many temples to the gods have been built by the 

 Athenians. 5. Let the door be shut at once. 6. Take care to have 

 consulted well before acting (lit., before the deed). 7. The desire of self- 

 government is implanted iu all men. 8. Let the robbers be slain at 

 once. 9. The enemy are said to have been shut up in the citadel. 10. 

 Xenophou's two sons, Gryllus and Diodorus, had been educated iu 

 Sparta. 



EXERCISE 89. ENGLISH-GREEK 



1. Ht<t>ovcvrcu. 2. Oi ?rat(5 iretj>ovtvina.i. 3. Oi CTTpanajTai eire(f>ovvvro. 



4. KaTaceK\67Tai. 5. KaraKCKXeo-fle. 6. KutreKeK\etaOe. 7. 

 fievoi eiatv. 8. 'i7 Avo avSptaira Kar(KK\ei<T9nv. 9. Oi /3oi/ K 



Keyoi-Tcu. 10. Eu ire7rai3uyuai. 11. Eu eveiraiSeuao. 12. Eu ircirau&evvTat. 

 13. Kaxwc eirenaibev/j.nv. 14. Ka/cwr eirejrai3ei/<ro. 15. Ta Sevipa eu iret, vrevrai. 

 16. To ievJpa KOKMC enetyvrevTO. 



K\ettrt)a 



* The difference between ei /3oi/Xei and eav /3uuXp may be thus ex- 

 plained: e< /3oi/Xct assumes that you are willing if you are willing, which 

 I believe you to be and so may be translated since ; eav pov\ri makes no 

 such assumption should you bt willing, about which I express no opinion 

 either way. 



Fig. 28. 



EXERCISES IN EUCLID. V. 



PROPOSITION XXIX. To trisect a given right angle ; that is, 

 to divide it into three equal parts. 



Let BAG (Fig. 28) be the given right angle ; in A c take any 

 point D, and on A r> describe an equilateral triangle A D E j 

 bisect the angle E A D by the line A F, B 

 meeting E D in F; then the lines A E, A F 

 will trisect the given right angle. For 

 since the three angles of a triangle are 

 together equal to two right angles, and 

 the angles of an equilateral triangle are 

 equal, the angle E A D is equal to one- 

 third of two right angles, i.e., to two- 

 thirds of one right angle. But the whole 

 BAG is a right angle ; therefore the re- 

 mainder B A E is equal to one-third of a 



right angle. And the angles E A F, FAD are each half the angle 

 E A D ; therefore each of them is one-third of a right angle ; 

 hence A E, A F trisect the right angle. Q. E. F 



PROPOSITION XXX. If two right-angled triangles have one 

 side and the base in the one equal to one side and the base in 



^ e other, each to 

 each, they shall be 

 equal in every respect. 

 Let ABC, DBF 

 (Fig. 29) be two tri- 

 angles, having ABC, 

 D E F right angles, and 

 let A B, A c in the one 

 be equal to D E, D F 

 in the other ; then 

 shall B c be equal to 

 E F, and the triangles 

 equal in every re- 

 spect. For if B c be not equal to E F, one of them must be 

 greater. Let E F be the greater, and from E F cut off E o = 

 B c ; join D G ; then since A B, B c are equal to D E, E G, each to 

 each, and the included right angles are equal, therefore base A c 

 is equal to baso D G (Euc. I. 4). But A C = D F by construction; 

 therefore D G = D F, and angle D G F = angle D F G (Euc. I. 5). 

 But since D E a is a right angle, D G E is less than a right angle 

 (Euc. I. 17) ; therefore D GF, which with DG E is equal to two 

 right angles (Euc. I. 13), is greater than a right angle. Hence 

 D G F and D F G together are greater than two right angles, 

 which is impossible by 

 Euc. I. 17 : hence E G is 

 not equal to B c. And 

 similarly it may be proved 

 that no line but E F is equal 

 to B c ; hence E F is equal 

 to B c, and the triangles 

 equal in every respect. 

 Q. E. D. 



Corollary. This propo- 

 sition is not necessarily true, as might be supposed, if the eqt 

 angle in the two triangles be not a right angle ; for in this ca 

 it is not necessarily untrue that D G is equal to D F. If 

 equal angle be not a right angle, there will be two positions 

 possible for the third side, as in Fig. 30, D F and D F' bot 

 satisfying the conditions of the proposition. This is, of cot 

 the " ambiguous case " of trig 

 nometry. 



PROPOSITION XXXI. The straight 

 lines which bisect the angles of 

 triangle meet in a point. 



Bisect the angles ABC, B c . 

 (Fig. 31) of the triangle A B c bj 

 the lines BG, CG meeting in G; 

 join A G ; then shall A G bisect tht 

 angle B A c. Draw G D, G E, G F perpendicular to A B, B c, c A ; 

 then in the two triangles G D B, G E B, since the right angle 

 G D B is equal to the right angle G E B, and by construction the 

 angle G B E is equal to the angle G B D, and the side G B is com- 

 mon ; therefore by Euc. I. 26 the triangles are equal, and ther 

 fore G D = G E. By an exactly similar course of reasoning G : 

 = G F, therefore G D = G F ; and because G D = G F, and G A ia 

 common, also right angle G D A = right angle GFA. therefor?, 



