LESSONS IN ALGEBRA. 



309 



glory. 15. Honour follow! diatinyuUhod virtue. 10. Ha proiniaed 



me thut hewouM ivtimi. 17. !!< luw n turned. 18. No, ho will return 



to-morrow. 19. BoyBittpportaaoh othar. -'>. lloya ouifht to support 



:, r. -Jl. 1 |.ii> and Hhull pity tlio wretched. 22. Never forget 



, fault*. S3. ] -t out within a fow dyn. Jl. Wli.-n 



wilt HI.IM r. turn ? .'.. lioya, ruverunoe the aged. 



Observe, that in the ablative absolute construction there are 

 properly tvvx> sentences, and consequently two subjects : for 

 rxiimple, sole oriente, tenebrce diffuyiunt ; in the words sole orient* 

 a subject, namely, sol; and in tenebrce diffuyiunt there 

 is a subject, namely, tenebrce. The former sentence is incom- 

 plete, nevertheless there is a subject in it. 



Now it is an element in the ablative absolute construction, 

 tint the subject of the sentence having the verb is not the 

 same as the subject of the imperfect sentence containing the 

 liartiriple. You may see this fact exemplified and illustrated 

 in these instances : 



ABLATIVE ABSOLUTE. 



1. Ssnescente lund, ostretc taboscere dicuntur. 

 The moon waning, oysters are said to waste away. 



2. GeryotM iiiferoiijito, Hercules in Italian) venit. 



mo inlo Italy. 



3. Sabini* debellatix, Tarquinius triumphans Bomam rediit. 



The Salines being subdued, Tarrjuiji tn triumph, returned to Rome. 



4. Chilo, flio tncfore OlyinpitD, preo gaudio exspiravit. 

 Chilo, his son BEINO conqueror at Olympia, died of joy. 



5. Apes, acitleo aininso, statim emori oxistimautur. 

 liees, their sting being lost, arc thought to die at once. 



It will be noticed from the fourth of the above examples, 

 that the participle, especially when it would be the participle 

 of the verb to be (which is not found in good Latin authors), 

 is sometimes omitted. 



KEY TO EXERCISES IN LESSONS IN LATIN. XXXII. 



EXERCISE 114. LATIN-ENGLISH. 



1. The safety of men depends not only on truth, but also on repu- 

 tation. 2. The citizens, having made a treaty with the enemies, 

 enjoyed peace. 3. By reflection, we comprehend God and the divine 

 mind. 4. We live on milk, flesh, and many other things. 5. Take 

 care that you do not avenge yourselves on your enemies. 6. The Bomaus 

 promised this to the Nurnidians. 7. The Numidians continued to 

 harass the Carthaginians by war. 8. The Bomaus are about to strive. 

 9. The Bomaus say that they will strive. 10. The Romans returned 

 the favour with increase. 11. The Romans promised the Numidians, 

 if they would continue to harass the Carthaginians by war, that they 

 would strive to return the favour with increase. 12. No one hns lived 

 too short a time who has performed a work of perfect virtue. 13. 

 Wise men despise the appearances in a dream. 14. As soon as we 

 have arisen, we despise the appearances in a dream. 15. Aristotle, 

 Zeno, and innumerable others, having gone out. of their country, never 

 returned home. 16. There is no plav^ue so detestable, which is not 

 produced by man against man. 17. I am not born for a corner. 18. 

 This whole world is my country. 19. The seeds of virtue are inborn 

 in our nature. 20. Hannibal fought with the Romans in Italy. 21. 

 Hannibal, having fought with the Romans, always came off conqueror. 

 22. Hannibal, as often as he fought with the Bomans in Italy, came 

 off conqueror. 



EXERCISE 115. LATIN-ENGLISH. 



1. The mind of every most excellent man chiefly strives after im- 

 mortal glory. 2. The enemies were wearied by the length of the con- 

 flict. 3. The enemies, wearied by the length of the conflict, left the 

 (field of) battle. 4. He acquired virtue. 5. In whatever part of the 

 world a good man is, he will be loved by friends. 6. He who has ac- 

 quired virtue, in whatever part of the world he is, will be loved by us. 

 7. Courage is eager for danger. 8. Courage does not reflect on what 

 it may be about to suffer. 9. Courage is eager for danger, and whither 

 it turns, does not think of what it will suffer. 10. Augustus did not 

 suffer himself to be called a lord. 11. Some animals nre destitute of 

 reason, others use reason. 12. The soul having escaped, the body is 

 worth nothing. 13. The memory of illustrious men, even when dead, 

 has influence with us. 14. It is worthy of a king to aid the fallen. 

 15. It is peculiar to folly to perceive the faults of others and to forget 

 its own. 16. We ought to do our best to benefit very many persons. 

 17. To be angry with those whom we out-lit to love is wickedness. 18. 

 Friendships, acquaintances, and neighbourhoods contain some pleasure 

 (something of pleasure). 19. We understand our advantages better by 

 being without them, than by enjoying them. 20. What pleasure 

 friendships, acquaintances, and neighbourhoods contain, we understand 

 better by being without them, than by enjoying them. 21. Fresh men 

 always succeeded wearied ones. 



EXERCISE 116. ENGLISH-LATIN. 



1. Felicitas virtute nititur. 2. Nititur ne hominibus felicitas? 3. 

 Non, felicitas Deo nititur. 4. Excolere virtutem eniti debemus. 5. 



Pater filium complex ua act. 0. Filiw patria mortem ultua act 7. 

 Bex premium pollioitus eat. 8. Sororl tiue regina polllciU eatna 

 premium t 9, MUitea cumulatam gloriam adipisci niteotur. ;o. 

 Mane experrecti aunt, at diaoeaaare. 11. Bena vital offlciia functi aunt. 

 12. Ariatotelea at Zeno pnaoaptorum offlcJU functi aunt, 13. Qoaado 

 amici tui domum revertent? 14. Heri domum roverterunt. \L. K 

 patria protect! aunt, et nunqoam revertent. 16. Peatis bjao bominum 

 in animia nata eat (bom). 17. Ubi eat patria? U. Patria ma* art 

 inuudua. 19. In animia morUlibua aunt aemina innat* vitiorum. 20. 

 Dux cum hoatibua congreaaua aat. 21. QuotieacunqiM ducea Anglici 

 cum hoatibua congress* aunt, aemper diaceaaere auperioraa. 22. Op- 

 tirni cujuaque pueri animus muxime parentea auoa amat. 23. Ikroi 

 in aalutem anitnco nituntur. 21. Lacta pnarl et poelto veacnntor. 0. 

 Diacipuli officiia auis functi aunt. 26. O Dana, miserere lapaoram. 

 27. Succurite pauperibua. 28. Proprium oat 3tultiti nulli prodeaae. 



LESSONS IN ALGEBRA. X. 



REDUCTION OF FRACTIONS. 



130. To REDUCE fractions of different denominators to fraction* 

 having a common denominator. 



Multiply togetlier each numerator and all the denominators 

 except its own, and the product vnll be the required numerator oj 

 each fraction ; next, multiply together all the denominators, and 

 the product will be tht required denominator of each fraction ; 

 these properly arranged in order will give the answer. 



EXAMPLE. Reduce - -, and to fractions having a com- 

 mon denominator. V 

 Here, a x d x y = ady, ~) 



c X b x y = bey, > are the three numerators, 

 and w X b X d = bdm, j 

 Also, 6 X d X y = bdy, is the common denominator. 



ady bey , bdm . 



Hence, the reduced fractions are , , , rr ana . A.ns. 



bdy bdy bdy 



The reason of this rule is plain, for the reduction consists in 

 multiplying the numerator and denominator of each fraction into 

 all the other denominators, a process which does not alter the 

 value of the fractions. [See Art. 121.] 



131. An integer and a fraction are easily reduced to fractions 

 having a common denominator, by making the former a fraction. 

 [See Art. 122.] 



Reduce a and to fractions having a common denominator. 

 c 



Here, a and -, are equal to - and -, which are equivalent to 

 c ic 



and - , the fractions having a common denominator. 

 EXERCISE 15. 



1. Reduce , and -to fractions having a common denominator. 



3m' g y 



2. Reduce -, -, and r - to fractions having a common denomi- 

 nator. 3 * d + h 



3. Reduce - , and to fractions having a common denomi- 



nator. 



4. Beduce a, b, - , and - to fractious having a common denominator. 



m y 



5. Reduce - -, and - to fractious having a common denominator. 



b d / 



6. Reduce , " , and - to fractions having a common denominator. 



a 56 2 



x c 



7. Reduce b, -, and to fractions having a common denominator. 



8. Reduce - , - -?. and to fractions having a common denominator. 



a i y 3 



9. Reduce - , , and x to fractions having a common denominator. 



a' 4c 5 



10. Reduce - , , --, and- to fractions having a common denominator. 



11. Reduce , 17, y , x, and c to fractions having a common de- 

 nominator. 



' to fractions having a common denominator. 



12. Reduce 



13. Reduce 

 denominator. 



14. Reduce 

 denominator. 



and - 



and 



. 

 - 1 



to fractions having a common 



. and to fractions having a 



-ox + ix* *+o 



