292 



THE POPULAR EDUCATOR. 



Atteso che, since. 

 A causa che, because. 

 Di maniera che, so that. 

 Eccetto che, unless. 

 In tanto che, so that. 



Mentre ciie, whilst. 

 Oltre che, besides that. 

 Secondo che, according as. 

 Se non che, except. 

 Subito che, as soon as, etc. 



In the following sentences will be found some examples of 

 this rule : 



Bisogua cotnpatirlo atteso che il 

 poverino e niatto in mezzo al 

 cervello, we mtist excuse him, 

 since the poor fellow is crack- 

 brained,. 



In tanto che non ritorno piii, 

 so that he returned no more. 



Subito che il sole f u levato, as soon 

 as the sun was risen. 



Rule 82. The following conjunctions require the verb which 

 follows them to be put in the subjunctive : 



Non ho cosa alcuna da dirvi se 

 non che conviene i vostri maes- 

 tri ubbidire, I have nothing to 

 say to you, except that you must 

 obey your masters. 



Di maniera che ella puo far cio 

 che la piace, so that you may do 

 what you please. 



Acciocche, that. 

 Affinche, in order that. 

 Ancorche, /hough. 

 Anzi che, be/ore that. 

 Avanti che, be/ore that. 

 Avveguache, though. 

 Benche, although. 

 Caso che, in, case that. 



Come che, althougli. 

 Come se, as if. 

 Con patto che, on con- 

 dition that. 



Contuttoche, although. 

 Dato che, suppose that, 

 lunauzi che, be/ore that. 

 Purche, provided that. 



Quando anche, al- 

 though. 



Quantuuque, though. 



Quasi, as if. 



Pognamo che, suppose 

 that. 



Prima che, before that. 



Seuza che, without that. 



The following sentences are examples of this rule : 

 Benche sia difficile, bisogna perd > Lo dice acciocche, non diate a me 



la colpa, he said it, that you may 



not lay the fault upon me. 

 Affinche ella gli scriva, in order 



that you may write to him. 

 Verro, purche non piova, f will 



come, provided it does not rain. 



vincere se stesso, however difficult 

 it may be, we wuist nevertheless 

 conquer ourselves. 



Ancorche sia in eta molto avan- 

 zata, nulladimeno gode perfetta 

 salute, he enjoys perfect health, 

 though in very advanced age. 



Rule 83. The following conjunctions sometimes govern the 

 indicative, and sometimes the subjunctive : 



Fino che, finche, finattantoche, ' Sebbene, though. 



infino che, infinche, mfinattan- ! Se bene, although. 



toche, till or until. Che, that. 



Perche, why. Conciosiache, conciosiacosache, 



Quando, when. for. 



Se,'/. 



In the following sentences will be found some examples of 

 this rule : 



Lo mio cuore non pub essere in 

 pace, finattantoche egli non si 

 riposi in voi, my heart cannot rest, 

 till it finds its repose in you. 



Che alcun non v' entrasse dentro, 

 infinattantoche egli tomato 

 fosse, that nobody should enter 

 until his return. 



Niuna doversi muovere del luogo 

 suo, finattautoche io noa ho la 

 mia novella finita, none of you 

 are to stir from your places, till 

 I put an end to my story. 



Sopra le rugiadose erbe, infinat- 

 tantoche alquauto il sole fu 

 alzato, colla sua compagnia 

 diportando se n' ando, she went 

 away, diverting herself v:ith her 

 company upon the dewy grass, 

 until the sun was a little higher. 



Chi te la fa, fagliele, e se tu non 

 puoi, tienlati a mente finche tu 

 possa, to him who plays you a 

 trick, play another, and if you 

 cannot, bear it in mind until you 

 can. 



LESSONS IN ALGEBRA. XLI. 



EVOLUTION OF COMPOUND QUANTITIES. 

 RULE. 1. Arrange the terms according to the powers of one 

 of the letters, so that the highest power shall stand first, the next 

 highest next, etc. 



2. Take the root of the first term, for the first term of the 

 required root. 



3. Subtract the power from the given quantity, and divide the 

 first term of the remainder by the first term of the root involved 

 to the next inferior poiver and multiplied by the index of the 

 given poiver ; the quotient will be the next term of the root. 



^ 4. Subtract the power of the terms already found from the 

 given quantity, and using che same divisor, proceed as before. 



PROOF. This rule verifies itself. For the root, whenever a 

 new term is added to it, is involved, for the purpose of sub- 

 Iracting its power from the given quantity ; and when the 

 power is equal to this quantity, it is evident the true, root is 

 found. 



EXAMPLE. 



Divisor A) 



Extract the cube root of 

 a 6 + 3a 5 - 3a - lla 3 + Qaf + 

 a 6 



3 a s _ 3a 4 _ lla s 



- 8 



Divisor B ) 6a 4 12a 3 + 6a 2 -f 12a - 8 

 3a*+6a3-3a 2 -6a + 4 _ Ga/4 _ 12a 3 , _ 8 



* * 



Divisor A is thus found, 3(a 2 ) 2 



3 X a X a* 



Sum = 3a 4 + 3a 3 + a 2 



Divisor B is thus found, 



3(a 2 +a) 2 



3 x ( - 2) X (a 2 -f a) 

 (-2)2 



Sum = 3a 4 + 6a 3 - 3a 2 - 6a + 4 



N.B. In finding the divisor in the 4th example of Exercis9 73, 

 the term 2a in the root is not involved, because the power next 

 below the square is the first power. 



The square root may be extracted by the following 

 EULE. 1. Arrange the terms of the given quantity according 

 to tJie powers of one of the letters, take the root of the first term, 

 for the first term of the required root, and subtract the power 

 from the given quantity. 



2. Bring down tivo other terms for a dividend. Divide by 

 double the root already found, and add the quotient both to tin: 

 root and to the divisor. Multiply the divisor, thus increased, 

 into the term last placed in the root, and subtract the product 

 from the dividend. 



3. Bring down two or three additional terms, and proceed as 

 before. 



PROOF. Multiply the root into itself, and if tlie product is 

 equal to the given quantity, the work is right. 



EXAMPLE. What is the square root of 



a 2 + 2ab + b 2 + 2ac + 2bc + c 2 (a + b + c 

 a 2 , the first subtrahend. 



2a + b) * 2ab -f V 



Into b = 2ab -f- ?> 2 , the second subtrahend. 



2a + 2b + c) 

 Into c = 



2ac + 2bc + c 2 [hend. 



2ac + 2bc -f c 2 , the third subtra- 



Proof. The square of the root a + b -f- c is equal to the given 

 quantity. 



For (a + b) 2 = a- + 2ab + b- = a ? -f (2a + b) X b. 



And substituting h = a -f- 6, the square h' = a 2 -}- (2a + b) X b 



And (a + b + c) 2 = (h + c) 2 = 7i 2 -f (27i + c) X c ; 

 that is, restoring the values of h and 7i 2 , 



(a +. b + c) 2 = o? + (2a + b) X b + (2a + 2b + c) X c. 



In the same manner it may be proved that, if another term 

 be added to the root, the power will be increased, by the pro- 

 duct of that term into itself, and into twice the sum of the 

 preceding terms. 



The demonstration will be substantially the same, if some 

 of the terms be negative. 



It will frequently facilitate the extraction of roots to consider 

 the index as composed of two or more factors. 



Thus a* = a* X ^. And a^ = a& X ^ That is, 



The fourth root is equal to the square root of the square root ; 



The sixth root is equal to the square root of the cube root ; 



The eighth root is equal to the square root of the fourth 

 root, etc. 



To find the sixth root, therefore, we may first extract the 

 cube root, and then the square root of that result. 



EXERCISE 73. 



1. Find the 4th root of a* + 8a 3 + 2-la 2 + 32a + 16. 



2. Find the 5th root of a 5 + 5a 4 b + 10a 3 !/ 2 H- 10a 3 b $ + 5ab + b 5 . 



3. Find the cube root of a 3 - 6a 2 b + 12ab 2 - 8b 3 . 



4 Find the square root of 4a 2 - 12ab + 9b + 16ah - 24lh + 16/1^. 



5. Find the square root of 1 - 4b + 4b 2 + 2y 4by + y 2 . 



6. Find the square root of a 6 -- 2a 5 + 3a 4 2a 3 -f a?, 



7. Find the square root of a* + 4a 2 b + 4b 2 - 4a 2 - 3& + 4. 



8. Find the square root of x* - 4a- 3 + 6x 2 - 4,r + 1. 



