Imágenes de páginas

I Tibb banks, offices ; stecchi (sték-kee),

**ris tsi-dá-skee), Germans; Turchi (toórr

- reese ; vecchio (vék-keco), an old man; 25. 437 Las feeáhn-kee), flanks, sides ; Ghe. Dei: çretto (ghét-to), a jewry; ghirlanda 3-2. pisi: quellino (ghi-bel-lée-no), Ghibellin; M-, sotels ; maghe (máh-gai), sorceresses ;

3-95-7*.loyments. us a sety arises for giving to the letter e before Le smi e in the word church, and to g before Le mani dy in ginger. Evidently a sign must be el Diaz is ez esse c would be sounded like k, and , e I = L 53 bis sign is the vowel i. In common enn **. searcely heard, serving the purpose only * se sus a sign; but in the more measured of

i raamastase the pulpit, the stage, public assemDorte Core te

a mi re se sentiy in the conversation of cultivated Dersom man

persons. Deze tocched in the enunciation, while the

Fa par pies to the pronunciation of the vowels a, o, Stocriter :

Lene izden

d. Eace sacche fundamental rule of Italian, which goes Sa. Sa zem. Iemes

site sh the coe above stated, that whenever a neces. Sy Sprieg to the e before a, 0, and u the compressed

zada - sie Enzish word church, and to g before a, o, and imran ne v Eseista Festes

e soprssed sound of g in ginger, the letter i (an auxiliary

ssz a sus casei must be placed between c and the vowels a, BIZSE :3 . I

2. sed , ad between , and the vowels a, 0, and u; and the 1. Eres the retas e bebe red cotons taas arising will be cia, cio, ciu, and gia, gio, gilt, song fuenteri uran be

Tronounced tebal, teho, tchoo, and jah, jo, joo). For example, menetren var mer :

ischah-skóo-no), everybody; ciancia (toháhn-tchab), sa bekeri 3. öc » se

El 2* i piery; so tedes), that, what; cioe (tchoê), that is to say: beef, watawat nå sitemize Frearers

Joico brakt-tehol, arm; ciuffo (tchóof-fo), I catch, I snap: mot hunder seram ng

so se ses 30 season-mah), a mob, a crew of galley-slaves ; giallo

it sored - yellow; giorno (jór-no), day; giudice (jóo-dee-tchai), 1. kata sa cum tenner, erthiesa se is. sender judge: giustizia (joo-stée-tzeeah), justice ; giubilo (jóo-bee-lo), heart ile in the mission 1 Egrevat joy jubilee. maoy the same lancer 2002 223 reti ca iyonite When e follows the letter s, thus forming the combination s6, However are us, werdens be senises nothing and when at the same time it precedes the vowels a, o, and 15. However we were a si v se bo. s.

or the consonants 1 and r, it will be clearly apparent that the Whatever may spe2 ha. Whatever in this case will follow the general rule, and be sounded like k, wad uk w KR keyin de temel . Whatever w troud Dugths to bas w se laserer be, weoty sea, seo, scu, scla, etc., scri, etc. (pronounced skah, sko, skog shall bear it 24. lubas se me as mura u buy read. 2. sklah, etc., skree, etc.). When, however, the combination Wo must love uvuky ay mayr s * Brez me these con immediately precedes the vowels e and i, the sound of the ci toas I au mua duwpe your post He sees eren fruiz less compressed than without the s before it; and sc in sud all the books tho mo Jesubasa

eases is sounded like sh in English words. The combination sce and sci will therefore be pronounced shai, or shê, and she

But when c, with a s before it and with e or i to follow, is t LESSONS IX TI-VI.

Ntain the sound of k just as before a, o, and u, recourse is he V PRONUNCIATION OF THB LETS EN COX to the same auxiliary letter h to indicate the preservation of th BINATION WITH OTHYS LSITSES

sound of e like k; and the combinations sche and schi are pe IT u usonstary that I should now ***) come degree of nounced skai, or skê, and skee. When, on the other hand, minuto ourtain peculiarities

or "tast for at oceurrere, with an s before it, and with the vowels a, 0, and u to follow, and consequently of the highest importa se promia to be pronounced not like skah, sko, skoo, but like sh, recours tom of the letters, y, and X, when they enter into certain com- must be had to the letter i, which is interposed between sc a. barmative with other letters

a, o, and 1, just as in those cases where, as we have seen, Willi regard to the letters and % I have a mai stated and standing by itself is to have the compressed sound of illustrated by examplow in the tirst pronouncing taon, that when church before a, 0, and u; and the combinations thus arisin 6. Wynio push bofore the vowels (m. v, and so is sounded scia, scio, and sciu, will be pronounced shah, sho, and sho lihat any like the English in the words yeupe, can, and guil. The previous observation holds good in this case likewise, th But one that it should be necessary in the declension of in more studied pronunciation the letter i is in these combin Hoe wha bonatim of verbs, etc., to give to the c and g tions slightly touched, though the voice must rapidly glide ir tin the wala a and I who wamo sound that c and g have the enunciation of the vowels a, 0, and u. Examples - Scar bellion, 49, Hit la obvious that somo sign must be used to (skáhrr-pah), shoe; scoppiare (skop-pêeah-rai), to burst, crae med that patiententiam of the r and 9, and avoid confusion. scuffia (skóof-feeah), a woman's cap; scherno (skérr-no), mooker The 17*** KA is the than the latter h, which, as has been re- schifare (skee-fáh-rai), to avoid, to have an aversion for; se

I lose the 1st in uriiniloem, written aign, and on that account mare (sklah-máh-rai), to exclaim; scrivere (skrée-vai-rai), to wri **** Hodt utest to the purpos. In this way we arrive at scelto (shél-to), selected; scevro (shái-vro), separated; scia

serwolves the main gh, and from what has been said it is (sháh-mai), a swarm of bees; coscia (kô-shah), thigh; scio dirbage truve le mont d ch before e and i can be no other (shól-to), ungirded ; sciocco (shok-ko), stupid ; asciutto (ah-aho 11 * 11,0 40r , and the mund of gh before e and i, that to), dry.

**, *** gone, go, and gull. And, indeed, The combinations gl, gn, and some others, I shall explain
** Italian grammar, that whenever a notes, as they occur in the next pronouncing table, wh
as in the inflections or terminational illustrates several combinations of the letters c, 9, and s.
sabing the sound of the e which in the
.4 A mand of g which in the root

moi qull, before the vowels e and i;

1. Che, Chi, Ghe, Ghi.
. mad, sad the towels e and i, and


xing # be 4, chi, Ard ghe, ghi


L; you give). For example, China


Descent, bent.




ahrr-tchê-rai Bowman, archer. Ghiro ghée-ro Dormouse.


ahrr-tchó-nai Saddle-bow, saddle. Rachele rah-ke-lai Rachel.


aht-tchóo-gah Anchovy. Archimede ahrr-kee-me-dai Archimedes.

No observation has yet been made in reference to the proVogherà vo-gai-ráh He will row.

nunciation of the double c (cc). This depends, as well as the Beghino

dai-ghée-no Biggin, a child's cap. Foche fô-kai

pronunciation of double 9 (99), on the vowel that follows the Sea-calves. fóe-kee

latter c. Fichi Fig-trees.

If that vowel is a, o, or u, the cc is sounded like a Leghe lái-gai Leagues, alliances.

double k (kk) or ck. For examble, bocca (bók-kah), mouth; becco Laghi láh-ghee Lakes.

(bêk-ko), beak; accusare (ahk-koo-záh-rai), to accuse. If, how. ever,

that vowel which follows the letter cis e or i, the double 2. Chia, Chie, Chio, Chiu. I have explained the combination chi to be sounded like kee. only perhaps stronger, and with vibration. On that account, I

C (cc) is sounded something like tch in the English word match, When one of the five vowels follows this syllable, it is so inti- have tried to imitate the stronger sound of the cc by the letters mately blended with the following vowel, that a kind of squeezed itch, placing the first t in the first syllable, and tch at the sound of chi is the result, the voice sliding, as it were, from beginning of the second, just as I have attempted to imitate cki to the next vowel with great rapidity.

the sound of the gg by placing d in one syllable and j at the Italian. Pronounced. English.

beginning of the next in such words as paggi (páhd-jee), pages, Chiaro keeah-ro Clear, bright.

attendants. The remark with respect to the pronunciation of Chiesa keeái-zah Church.

the gg, however, holds good of cc; the voice must not pause too Chiodo keeó-do Nail, I nail.

long on the t of the syllable where the first c occurs, and glide Chiuso keeóo-zo Inclosed, inclosure.

as quickly as possible to the pronunciation of the second c, Tarchiato tahrr-keeáh-to Plump, fat.

which must be very much vibrated. In this way a more equal Banchiere

bahn-keee-rai Banker. Melchiorre mel-kecorr-rai Melchior.

distribution of the sound tch between the two syllables will be Conchino kon-keeóo-zo Concluded.

effected, which will produce the correct sound of the cc; and Vecchia vék-keeah An old woman,

my imitation of that sound by ttch has no other object than to Secchie sék-keeai Buckets.

indicate to the reader the necessity of giving a stronger vibraSecchio soók-keeo Auger, juice.

tion to the cc. It is obvious that when cc is followed by consoSckiuma skeeóc-mah Froth, scum.

nants, it must be pronounced like k, just as the single c in the 3. Gia, Glie, Glio.

like case must be so pronounced. For example, acclamare (ahk. The remark made with respect to the syllable chi, followed klah-máh-rai), to elect by acclamation, to applaud; accrescere by any of the five vowels, is equally applicable to the syllable (ahk-krái-shai-rai), to increase, etc. When between the cc and ghi followed by a vowel : here, likewise, the syllable ghi is, as it the yowels e or i the letter h is interposed, the cc is also sounded Fere, squeezed, and the voice must slide into the pronunciation like k, as well as the single c in such cases and for the same of the vowels that follow ghi with great rapidity.

reasons, the h being a mere auxiliary letter to indicate that cc

before e and i is not to have the sound of ttch, but of kk, as in Italian. Pronounced. English.

chicchera (kík-kai-rah), a tea-cup; chiacchiera (keeáhk-keeai-rah), Ghiaja gheeák-yah Gravel, sand.

A ferrule.


Clod, turf.


rin-gheenh-rai To snarl.

iu jóo-sto

prai-ghecê-rah Prayer, desire.


páhn-tchah Belly, paunch. Singhiozzo sin-gheeó-tso Sob, sigh, hiccough.


Kind, species.
The double zz, as well as the single m, may have the mild Lercio


Dirty, foul. sound of the word adze (with which, by the bye, the ds in the Ciufto


I catch or snap. word Windsor corresponds), or the hard sound of tz in Switzer- Regia


Royal palace. burd. According to modern orthography, the letter s is gene



A Roman coin, July. n's doubled in the middle of words between two vowels, and the pronunciation of this zz scarcely differs from that of the

5. Gua, Gue, Gui, Guo; Qua, Que, Qui, Quo. single z. However, before diphthongs—as, for example, ia, ie, Italian,


English, and i must remain single, and has always, in such a case,


A ford. the sharp sound. For example, ringraziare (rin-grah-tseeáh-rai),


A Guelph, an ancient coin

of Florence. to thank ; pigrizia (pee-grée-teeeah), idleness; inezie (ee-ne.


Leader, guide. trezzi), follies; Bonifazio (Bo-nee-fáh-tseeo), Boniface.


I follow or pursue.
thín-gheeah Girth.


Almost, as if.
oón-gheeai Nails, hoofs.


I grin, grinding the teeth. Quito.


I acquit, free. 4. Cia, Cie, Cii, Cio, Ciu; Gia, Gie, Gii, Gio, Giu.


Leather, skin.

sai-gwah-tchai Follower, disciple.


in-sé-gwai He pursues. Cinno tchah-no Blue-bottle (plant).


in-gwee-nai Groin. Cieng tchái-rah The look, face.


lee-qwó-ree Liquors. The vowel i before e, when both follow the consonant c, is


ah-kwah.reeo Aquarius. Loquela

lo-kwê-lah Speech. cacanced as though the i was not there, and the whole com


áh-qwee-lah Eagle. tration only ce. The same remark, however, made with regard


ah.qwó-80 Aqueous, watery. so the combinations cia, cio, and ciu—that in a more measured Lingua


Tongue, language. Eunciation the vowel i in these cases is slightly touched-holds Sangue


Blood. good here also.


Fat, plump.

Pasqua Ciofo

Easter. tchó-fo

A mean fellow.


An ass.



Unjust, iniquitous,



Thou comparest.
Ice, frost, cold.
The observation just made in the foregoing note with respect

6. Cla, Cle, Cli, Clo, Clu; Gla, Gle, Gli, Glo, Glu. to sie is strictly applicable to the syllable gie. It is always pro



Lounced as though the i was not there, unless slightly touched


kle-ro is measured pronunciation.


jo vai
Jove, Jupiter.


Clotho, one of the Fates.


Clusium, a town.
bah-tcháh-rai To kiss, salute.


Sword, poniard.

2. aa

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- al + bd

This is the first occurrence in these lessons of the important on the principle that the product of the divisor into the several combination gl. It has two different sounds. When it is not parts of the quotient is equal to the dividend. [Art, 92.] followed by the letter i it has the sound of gl in gland, glebe, 107. Rule.--Arrange the terms so that the letter which is in the giory, glue ; and this sound can offer no difficulty. But when first term of the divisor shall also be in the first torm of the the combination gl is followed by the letter i and one of the dividend. If this letter is repeated as a factor, either in the vowels a, e, 0, and u, it is pronounced precisely as the double l divisor or dividend, or in both, the terms should be arranged in (il) in the French words bouilli, fille, gresiller, grenouille, bouillon, the following order : put that term first which contains this letter bilard, billet, brouillon, fevillu, and, generally speaking, in all the greatest number of times as a factor ; then the term containing those words where the Il has after the vowel i a squeezed sound it the next greatest number of times, and so on. in the French language. They who are unacquainted with EXAMPLE.—Divide 2aab + bbb + 2abb + aaa by aa + bb + ab. French may form a notion of this sound by separating and in- If we take aa for the first term of the divisor, the other terms verting the gl in the enunciation, that is, by pronouncing u must be arranged according to the number of times a is repeated s before the g, and changing the latter into y. Only the first l as a factor in each. Thusmust go to one syllable, and the second l along with the y, and

Dividend. with a squeezed sound to the beginning of the next, while care Divisor aa + ab +66 ) aaa + 2aab + 2abb + bbb (+6 Quotient. most be taken that the voice should glide rapidly from one syl.

oaa + aab + abb Lable to the other, by which means a more equal distribution of the squeezed sound lly will be produced, and a correct pro

gab + abb + bbb nunciation of the gl effected. An approximation to this sound

aah + abb + bbb may be found in the English words million, miliary, biliary, billarda, seraglio, intaglio, and oglio. The letter i, between the In division, it is necessary that the strictest attention be paid combination gi and the vowels a, e, o, and u, is (as well as in to the rules for the signs in subtraction, multiplication, and the combinations cia, cio, ciu, and gia, gio, giu) a mere auxiliary


EXERCISE 8. letter, i.e., & mere soundless written sign, to indicate that gl before a, e, o, and u is not to have the sound of gl in gland, Perform the following exercises in division globe, glory, and glue, but that squeezed sound, the imitation and 1. 2x - 2xy + yy + r - y. description of which I have here attempted.

bb = a + b. Gleba

3. bb + 2bc + cc + b + c.
Clod of earth.

4. aaa + xXx + a + .
Glyph (in architecture).

5. Par

Saaxy + 60aax + ary – 2y + 2a - y.

6, a +b--ar- bz + cm a + b - c.

7. ac + bc + ad + bd + 2a + b. Reclamo rai-klah-mo Reclamation.

8. ad

bh + y d-h. For example : voglio (váhl-lyo), a siove; meglio (mêl-lyo), better; piglio píl-lyo), I take, seize; miscuglio (mis-kóol-lyo),

108. From the preceding principles and examples we derini mixture ; svegliare (zvel-lyáh-rai), to awake; togliere (tal-lyai the following rai), to take away ; scegliere (shél-lyai-rai), to choose ; doglia

GENERAL EULES FOR DIVISION. (dol-lyah), sorrows; bigliardo (bil-lyáhrr-do), billiards ; biglietto

(1.) Division, in all cases, may be expressed by writing the (bil-lyét-to), note, bill ; imbroglione (im-brol-lyó-nai), a meddling divisor under the dividend in the form of a fraction. fellow; fogliuto (fol-lyoó-to), full of leaves. Egli, he; eglino,

(2.) When the divisor and dividend are both simple quantitie they; quegli, that one; gli (the plural of the article or the pro- and have letters or factors common to each : divide the o noun), with its numerous compositions, and gli, the final inflec. efficient of the divisor by that of the dividend, and cancel tion or terminational syllable of nouns and verbs, have always factors in the dividend which are equal to those in the divisor the squeezed sound Ulyee; while the mere syllable gli, at the

(3.) When the divisor is a simple, and the dividend a commencement and in the middle of words, always has the pound quantity: divide each term of the dividend by : sound of gl in gland, glebe, etc. The only exception is Angli, divisor as before ; setting down those terms which cannot Englishmen, (pronounced áhn-glee). For example: figli (fil-lyee), divided in the form of a fraction. Sou; fogli (fol-lyce), leaves of paper; gigli (j1-lyee), lilies ;

.(4.) If the divisor and dividend are both compound quantità weybjore (nai-gle-jai-rai), to neglect; negligente (nai-glee-jên-te), arrange the terms according to Art. 107. mogligent; negligenza (nai-glee-jôn-tsah), negligence; negligentare (5.) To obtain the first term in the quotient, divide the i (nai-give-jen-táb-rai), to neglect.

term of the dividend by the first term of the divisor. Multi the whole divisor by the term placed in the quotient; subtr

the product from the dividend ; and to the remainder bi LESSONS IN ALGEBRA.-VIII. down as many of the following terms as shall be necessary DIVISION (continued).

continue the operation. Divide again by the first term of

divisor, and proceed as before, till all the terms of the divid DIVISION BY COMPOUND DIVISORS.

are brought down. If the signs in the divisor and dividend Art 106, continued], --EXAMPLE.--Divide ac + bc + ad + bd, alike, the quotient will be +; if unlike, the quotient will be by oth Here, arranging the quantities for division as we do in com.

EXERCISE 9. Hay arithmetie, we have

1. Divide 12aby + 6abx - 18bbm + 246 by 6b.

2. Divide 16a 12 + 8y + 4 - 20ad.x + m by 4. Danimar a + b ) oct be + ad + 1d (c + d Quotient.

3. Divide (a – 2h) R (3m + y) * = by (a - 2h) * (3m + y). on be, the first subtrahend.

4. Divide and - dad + 3ay - a by hd - 4d + 3y

5. Divide ar – ry + ad - Amy - 6 + a by - a. * ad. + d.

6. Divide amy + 3my muy + am - d by

dmy. od. + bd, the second subtrahená.

- 6 + 2r - hd + 6 by Pard.

8. Divide bax - 8 + 2xy + 4 - Chy by 4axy. He se ma, tiba hrat torn of the dividend, divided by a, the first

9. Divide 16abcæ – 12xyab + 2 tabad 36ahgo by ab. ter tave divimur (Art. 92), gives c for the first term of the

10. Divide 21aaby + 42cdzaa + 14aaa 35aanab by 7an. utant Multiplying the whole divisor by this term, we have 11. Divide 12abxyz - 6hdabæy + 24xyabm by 3abxy. 4 budunt u + be, which is to be subtracted from the two first 12. Divide 3ax - 36bx + 42 - 72cx + 30ax by 3r. in the dividend. The two remaining terms are then 13. Divide 40ab – 4 (x + y) + 72 + 12 (a + b) + 480 by

14, Divide abz cdx + 8gx + x by ab laght down, as iu arithmetical division, and the first of these

- cd + 8g . hret torn of the divisor, as before, gives d for the

15. Divide 24xys – 36cd - 48abcd by 12xys - 18cd 24abed,

16. Divide ab - ad + ax (a + b) - laxy + ab by # quotient. Then multiplying the whole

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

2am. the product ad. + bd, which is to be sub

18. Divide xyz + 6x + 2z - 1 + 2xyz (a + b) by Gxyz. lining term of the dividend ; as no re

19, Divide - Bac - 12bc - bab - 10 – 2aabbce by Cabe. ision is complete.

20. Divide 18abyæ + 16aba 20bbom + 24ub by 25. rusta the following rule, which is founded 21. Divide 16x - 24 + 8 + 84 - 20ar - a by -4.



7. Divide ard

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16 by y

+ 20%.

22. Divide (* – y) * (38 + x) * b by (x - y) * (3a + x).

Here, 6a: + 7ax 3x2 ) 6a2 + Ilaz + 3x2 (1
23. Divide 11d (4- a) * (x + y) by (4 – 2) × 410.

6a? + 7aux 3.12
34. Divide - 40xy + 7abx
3ahmx by - 40y + 7ab 3ahm.

4ax + 6x2 Remainder.
25. Divide 20 (ab + 1) – 60 (ab + 1) + 50 (ab + 1) by 5a.
%6. Divide bax + 2xy 3ab by + 3ac + cy + h by 3a + y.

Now dividing this remainder by 2x, we have 2a + 3x for the
— ва
27. Divide aab - 300 + lab

4b + 12 by b

next divisor. %. Divide bb + 3bc + 2ce by b + c.

Divisor. Dividend. Quotient. 99. Divide Sanab - Ubbb by Pab bb.

2a + 3x ) 6a? + 7ax 322 ( 3a -
30. Divide za - 3arx + 3aas aaa by -a.

6a2 + ax
Jl. Divide ?yyy - 19yy + 20y
39. Divide zuzama - 1 by : -1.

2ax 3.12
3. Diride bazax - 9.02 + 6x – 3 by 2x + 3x – 1.

2as 302 The preceding rule may be thus summed up :-Divide every

The first remainder was divided by 2x because it is a common part or term of the dividend by the whole divisor, and collect factor of both terms of that remainder, and it cannot form a the results as in addition; the sum will be the quotient.

factor of the common measure, not being a factor of every term EXERCISE 10.

in the proposed quantities. As the division of the preceding

divisor by this simplified remainder leaves no remainder, there-
1. Divide cb2c3 by abc, and x@yu by z’y.

fore 2a + 3x is the common measure required.
2. Divide 20 + 1 and xb-1 each by .
3. Divide cityt - 4473 + afz' by 133".

4. Divide 2 – 3aty – 6a3ya by 3a2.

1. Find the greatest common measure of 28 - br and x2 + 2br + b2. 3. Divide 3asks – 102*6* + 8a365 by 32342 4a3.

2. Find the greatest common measure of cx + ad and a'c + a e. 8. Dinde – 13423 + 12a*rby 22 + 3ax 4a2.

3. Find the greatest common measure of 376 — 24x – 9 and 21:3 7. Divide - Ystyø + 12y3 – 4y by - 3xy + 212.

- 16% - 6. 8. Divide » - 6x + 522 + 12x + 4 by zde - 3x - 2.

4. Find the greatest coinmon measure of a*. 64 and 25 - bla. 9. Divide z* + cby 2 - a.

5. Find the groatest common measure of co? 1 and xy + y. li. Divide q? - 62 + 2bc ca by a + b - c.

6. Find the greatest common measure of 2% 23 and 2* -a'.
11. Divide 8lz: + 34x by 3x3 + 2t.

7. Find the greatest common measure of a? ab 26* and a? - 3ab
12. Divide + c3 by + a?.
13. Diride 10ye Pays + 4aạyk by 5y– tay3 + a'y?.

8. Find the greatest common measure of at - x and a3 - ai - az?
14. Divide 724 Be + 50.23 – 74.1 + 35 by x - 3cm + 5x - 7.

+ 15. Divide 2 3x3y + 2z+ya + yo by r


9. Find the greatest common measure of a3 – ab? and a? + 2ab + b2.
16. Divide z' by za + 2x + 1.
17. Divide & by za 2x + 1.

18. Divide ** -- 8x + 7 by ** - 3x + 2.

114. A common multiple of two or more quantities is a 19. Divide 5ct – 60x3 - 203x – a' by ** – 2x + a. 30. Divide ze - 3:49 + 3z** – po by 3

quantity which can be divided by each of them without a


- 3r T 3.00 22. Divide 33 – 37** + 35x3 + 729 + 2 by 2 + 309

remainder. Thus 12ab is a common multiple of 40 and 6b ; or

Ar - 2.
22. Divide jab + 9a'be 4ab3 + 4b3c - θαύc 96c3 by 3a - 20 + 3c.

of 3a and 2b, eto.
23. Divide a3 + 30% + 3ab* + 273 + 38°c + 3bc2 + c3 by a + 2b + c. 115. The least common multiple of two or more quantities is
84. Divide 4,5 - 2 + 4x by 2x2 + 3x + 2.

the least quantity which can be divided by each of them without 35. Divide 1 - 97 + 8x10 by 1 - 2x + x2.

a remainder. Thus 12abc is the least common multiple of 4a,


116. To find the least common multiple of two or more given 109. A common

measure of two or more quantities is a quantities. antity which will divide or measure each of them without Rule.-—

Reduce the given quantities to their prime factors ; find & remainder. [Art. 30.] Thus 2d is a common measure of the product of the greatest powers of these factors, and it will be 134, 64, 8d, etc.

the least common multiple required. 110. The greatest common measure of two or more quantities EXAMPLE. - Find the least common multiple of (a + x)?, E the greatest quantity which will divide these quantities with a? --- ??, and (a — «)?. cot a remainder.

Thus 6d is the greatest common measure of Here, the prime factors of the quantities are (a + a), (a + r), 124 and 18d; and 8 is the greatest commou measure of 16, 24, (a ---), and (a — x)2; now of these factors, which are different

powers of a ta and a-- a, the first and last contain their 111. To find the greatest common measure of two given highest powers; therefore, according to rule, (a + )(a — x)?

= (a? — 2a) is the least common multiple of the quantities
Bul.-Diride the greater of the given quantities by the less, required.
the decisor by the remainder, and every successive divisor by its

Para remainder, until nothing remains; the last divisor will be the 1. Find the least common multiple of be, ce, and bg.
Teatest common measure.

2. Find the least common multiple of a48 and asb?.
112. To find the greatest common measure of three or more 3. Find the least common multiple of 2ab, 3bc, Acd, 5de, and bef.

4. Find the least common multiple of (a + b)?, (a? – 6), (a - b)", Pal.- Find the greatest common measure of any two of them ;

and (a - b). ther the greatest common measure of that one and another of the

5. Find the least common multiple of 62, 9a?, and 4a2.

and and a? – m.

6. Find the least common multiple of 63 –
pantities, and so on, tiil all the quantities have been employed in
the yperation; the last divisor is the greatest common measure.

7. Find the least common multiple of (x -a), (x + a), (2 – a?), and

(2+ a2).
The greatest common measure of two quantities is not altered
by multiplying or dividing either of them by any quantity which

sot a divisor of the other, and which contains no factor which KEY TO EXERCISES IN LESSONS IN ALGEBRA.
ke i titisor of the other.

Te common measure of ab and ac is a. If either be multi-

7. - m

Ited by the common measure of abd and ac, or ab and acd,
On the other hand, if ab and acd are the given

hm - 3y
sities, the common measure is a; and if acd be divided by


8. y + the connon measure of ab and ac is a.

9. 1. 113. Hence, in finding the common measure by division, the

10. 1. Griect may often be rendered more simple by dividing it by

11. 1.

12. 6 +1. quantity which does not contain a divisor of the dividend.


13. 6-1. the vidend may be multiplied by a factor, which does not

14. xy -1 + 24.

5. dy +r-
uztain measure of the divisor.

15. ab + 1-2m.
EZAPLE.—Find the greatest common measure of ba? + 11ax
+382 and 6a? + 7 as

6. 2h + d +


od 32





a +r


Y am



16. 5.

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The words which are to be sung to the following tune were Wordsworth. Southey was a very voluminous writer, and 23 written by Robert Southey, the poet-laureate, who succeeded the good a biographer as he was a poet; his accounts of the lives Rev. Henry James Pye in this office in 1813, and held it till his of Nelson and Wolsey being inferior to no similar works in the death in 1843; when he was succeeded, in his turn, by William English language.

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2. And when the evening light decays,

And all is calm around,

There is sweet music in his ear

The sweetest is the voice of love
From distant sheep-bell's sound.

That welcomes his return. Our students must bear in mind that the exercises pe are of these notes are effectively illustrated here, at the opening of now giving are illustrative of the effect of TE and RAY. Both the second verse.

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