246 



THE POPULAE EDUCATOE. 



This is the first occurrence in these lessons of the important 

 combination gl. It has two different sounds. When it is not 

 followed by the letter i it has the sound of gl in gland, glebe, 

 glory, glue ; and this sound can offer no difficulty. But when 

 the combination gl is followed by the letter i and one of the 

 vowels a, e, o, and u, it is pronounced precisely as the double I 

 (II) in the French words bouilli, fille, gresiller, grenouille, bouillon, 

 billard, billet, brouillon, feuillu, and, generally speaking, in all 

 those words where the II has after the vowel i a squeezed sound 

 in the French language. They who are unacquainted with 

 French may form a notion of this sound by separating and in- 

 verting the gl in the enunciation, that is, by pronouncing II 

 before the g, and changing the latter into y. Only the first I 

 must go to one syllable, and the second I along with the y, and 

 with a squeezed sound to the beginning of the next, while care 

 must be taken that the voice should glide rapidly from one syl- 

 lable to the other, by which means a more equal distribution of 

 the squeezed sound lly will be produced, and a correct pro- 

 nunciation of the gl effected. An approximation to this sound 

 may be found in the English words million, miliary, biliary, bil- 

 liards, seraglio, intaglio, and oglio. The letter i, between the 

 combination gl and the vowels a, e, o, and u, is (as well as in 

 the combinations cia, do, civ,, and gia, gio, giu) a mere auxiliary 

 letter, i.e., a mere soundless written sign, to indicate that gl 

 before a, e, o, and u is not to have the sound of g I in gland, 

 glebe, glory, and glue, but that squeezed sound, the imitation and 

 description of which I have here attempted. 



GZeba gle-bab. Clod of earth. 



Gfli/o gli-fo Glyph (in architecture). 



Globo glo-be> Globe. 



Gluma gloo-mah Chaff. 



.Reclame rai-klah-mo Declamation. 



For example : vaglio (vahl-lyo), a sieve ; meglio (mel-lyo), 

 better; piglio pfl-lyo), I take, seize; miscuglio (mis-kdol-lyo), 

 mixture ; svegliare (zvel-lyah-rai), to awake ; togliere (t61-lyai- 

 rai), to take away; scegliere (shel-lyai-rai), to choose; doglia, 

 (dol-lyah), sorrows ; bigliardo (bil-lyahrr-do), billiards ; biglietto 

 (bil-lyet-to), note, bill ; imbroglione (im-brol-lyd-nai), a meddling 

 fellow; fogliuto (fol-lyod-to), full of leaves. Egli, he; eglino, 

 they ; quegli, that one ; gli (the plural of the article or the pro- 

 noun), with its numerous compositions, and gli, the final inflec- 

 tion or terminational syllable of nouns and verbs, have always 

 the squeezed sound llyee ; while the mere syllable gli, at the 

 commencement and in the middle of words, always has the 

 sound of gl in gland, glebe, etc. The only exception is Angli, 

 Englishmen, (pronounced ahn-glee). For example : ^fZi(fil-lyee), 

 sons; fogli (fol-lyee), leaves of paper; gigli (jfl-lyee), lilies; 

 negligere (nai-glec-jai-rai), to neglect ; negligente (nai-glee-jen-te), 

 negligent; negligenza (nai-glee-jen-tsah), negligence; negligentare 

 (uai-glee-jen-tah-rai), to neglect. 



LESSONS IN ALGEBRA. VIII. 



DIVISION (continued). 

 DIVISION BY COMPOUND DIVISORS. 



[Art. 106, continued"]. EXAMPLE. Divide ac + be + ad + bd, 

 by a + b. 



Here, arranging the quantities for division as we do in com- 

 mon arithmetic, we have 



Divisor a + b)ac + bc-\-ad-}-bd(c-{-d Quotient. 

 ac + be, the first subtrahend. 



* * ad + bd 



ad + fed, the second subtrahend. 



Here ac, the first term of the dividend, divided by a, the first 

 term of the divisor [Art. 92], gives c for the first term of the 

 quotient. Multiplying the whole divisor by this term, we have 

 the product ac -\- be, which is to "be subtracted from the two first 

 terms of the dividend. The two remaining terms are then 

 brought down, as in arithmetical division, and the first of these 

 divided by the first term of the divisor, as before, gives d for the 

 second term of the quotient. Then multiplying the whole 

 divisor by d, we have the product ad + bd, which is to be sub- 

 tracted from the remaining term of the dividend ; as no re- 

 mainder is left, the division is complete. 



This operation suggests the following rule, which is founded 



on the principle that the product of the divisor into the several 

 parts of the quotient is equal to the dividend. [Art. 92.] 



107. Rule. Arrange the terms so that the letter which is in the 

 first term of the divisor shall also be in the first term of the 

 dividend. If this letter is repeated as a factor, either in the 

 divisor or dividend, or in both, the terms should be arranged in 

 the following order : put that term first which contains this letter 

 the greatest number of times as a factor ; then the term containing 

 it the next greatest number of times, and so on. 



EXAMPLE. Divide 2aab + bbb + 2abb -f aaaloyaa + bb + ab. 



If we take aa for the first term of the divisor, the other terms 

 must be arranged according to the number of times a is repeated 

 as a factor in each. Thus 



Dividend. 



Divisor aa + db + bb ) aaa -f- 2aab -f 2afefe + febfe ( a -f b Quotient 

 aaa -j- aab -j- abb 



aab -f abb + bbb 

 aafc-f- abb -{-bbb 



In division, it is necessary that the strictest attention be paid 

 to the rules for the signs in subtraction, multiplication, and 

 division. 



EXERCISE 8. 



Perform the following exercises in division : 



1. xx 2xy + yy x y. 



2. aa bb -r a + b. 



3. bb + 2bc + cc * b + c. 



4. aaa + xxx * a + x. 



5. 2ax 2aax Saaxy + Saaax + axy xy -f- 2a y. 



6. a -I- b c ax bx + ex + a + 6 c. 



7. ac + be + ad + bd + x * a + b. 



8. ad - ah + Id - bh + y -H d - h. 



108. From the preceding principles and examples we derive 

 the following 



GENERAL RULES FOR DIVISION. 



(1.) Division, in all cases, may be expressed by writing the 

 divisor under the dividend in the form of a fraction. 



(2.) When the divisor and dividend are both simple quantities, 

 and have letters or factors common to each : divide the co- 

 efficient of the divisor by that of the dividend, and cancel the 

 factors in the dividend which are equal to those in the divisor. 



(3.) When the divisor is a simple, and the dividend a com- 

 pound quantity : divide each term of the dividend by the 

 divisor as before ; setting down those terms which cannot be 

 divided in the form of a fraction. 



(4.) If the divisor and dividend are both compound quantities, 

 arrange the terms according to Art. 107. 



(5.) To obtain the first term in the quotient, divide the first 

 term of the dividend by the first term of the divisor. Multiply 

 the whole divisor by the term placed in the quotient ; subtract 

 the product from the dividend ; and to the remainder bring 

 down as many of the following terms as shall be necessary to 

 continue the operation. Divide again by the first term of the 

 divisor, and proceed as before, till all the terms of the dividend 

 are brought down. If the signs in the divisor and dividend are 

 alike, the quotient will be + ; if unlike, the quotient will be . 



EXERCISE 9. 



1. Divide 12aby + 6abz - 18bbm + 24b by 6b. 



2. Divide 16a - 12 + 8y + 4 - 20adx + m by 4. 



3. Divide (a - 27i) x (3m + y) x x by (a - 27i) x (3m + y). 

 4,. Divide ahd - 4ad + Say - a by Tid - 4d + 3y 1. 



5. Divide.aa; ry + ad 4my 6 + a by a. 



6 . Divide amy + 3my mxy + am d by dmy. 



7. Divide ard 6a + 2r hd + 6 by 2ard. 



8. Divide 6ax - 8 + 2xy + 4 - 6hy by 4axy. 



9. Divide 16abca; 12a;yab + 24abd 36a7igb by 4ab. 



10. Divide 21aaby + 42cda;aa + 14aaa 35aaaab by 7aa. 



11. Divide 12abzyz Ghdabxy + 24xyabm by Sabxy. 



12. Divide 3aa; - S6bx + 42 - 72ca; + SOax by Zx. 



13. Divide 40ab - 4 (x + y) + 72 + 12 (a + b) + 48c by - 4. 



14. Divide aba; cdx + 8gx + x by ob cd + 8g + 1. 



15. Divide 24yz - 36c<J - 48abcd by I2xyz IScd 24abccJ. 



16. Divide ab ad + ax (a + b) 42a:y + ab by a. 



17. Divide 6am - 10a7i + 20 - 12cd + 17a by - 2am. 



18. Divide xyz + 6x + 2z - 1 + 2a;yz (a + b) by 6xyz. 



19. Divide - 6ac 12bc 6ab 10 - 2aabbcc by - 6abc. 



20. Divide ISabyx + I6abx 20bbcm + 24ab by 26. 



21. Divide I6x - 24 + Sa + 43 - 20a - a by - 4. 



