LESSONS IN ALGEBRA. 





thing, and capable of being employed in a paarive form, which 

 IB not the oaae with euere, though thin in the auxiliary, by 

 the aid of which the paaiive form of all Italian verb* u conju 

 pitted, 



In ettere the participle *tato change* with the gender and 

 number of the person or thing to which it re/en; a* 

 (uo-wo) xu-iw std-to, i-o (<(.'m-/iu) t6-no td-ta, noi (md-chi, 

 nmloH) *tu-mo std-ti, v6i (ffm-mi-ne, female*) $i-te ttd-te, etc. 

 : also the case with all positive, intransitive, and reci- 

 procal verbs, which are conjugated by moans of the auxiliary 

 ettere. In tho compound tenses of avert, on the contrary, the 

 participle avuto does not change its number and gender : for 

 example i-o (uo-mo) M a-vu-to, fu (don-na) ha-i a-vu-to, tun 

 (mA-tchi) ab-bid-mo a-vu-to, v6i (fem-mi-ne) a-v^-te a-vu-to. 

 This is also generally tho case with the pant participlen of all 

 active verbs which are conjugated by the aid of avere. Some- 

 times, however, tho participles of active verbs must adopt the 

 number and gender of the verb to which they refer, which will 

 bo explained later. 



the negative form of tho auxiliaries arcrc and esserc, 

 oil" I indeed of all other verbs, non is used, and always placed 

 before the verb ; as 



Non essendo state or state, not 



Ji'irt'ii'j been. 

 Io non ho, tu non hai. etc., 7 hare 



not, thou hast not, etc. 

 lo non ho avuto, I have not had, etc. 

 lo non sono, tu non sei, etc., 1 am 



not, (/ion art not, etc. 

 lo nou sono state or state, I have 



not been, etc. 



Non avere, no' (<> haw. 



Nou essere, not to be. 



Non aTendo, not h . 



Non essendo, not being. 



Non avero avuto, not to have 



had. 

 Non essore steto or steta, not to 



have been. 

 Non aveudo avuto, not having liad. 



In tlio interrogative form tho personal pronouns are of course 

 put after the verb, and when omitted the inflection of the voice 

 must, to some extent, supply their place ; as 



Ho io ? ?iaoe I f 

 Hai tu ? hoet thou f 

 Ha egli ? hat he f 

 Abbiamo noi ? have we f 

 Avete voi ? have you ? 

 Hanno eglino ? hare they? 

 Ho io avuto ? have I had f etc. 



Sei tu? art thouf 

 egli ? it her 

 Siamo noi ? are toe f 

 Biete voi P are you ? 

 Souo eglino P are then T 

 Sono io stato or steta? have I 

 been ? etc. 



Sono io ? am 7 f 



In the negative and interrogative form, the auxiliary or (in 

 simple tenses) tho verb is placed between the negation and the 

 pronoun ; for example 



Non eono io ? am 1 not f \ Non ho io ? Juice I not f 



The phrases there is, there are, there was, there were, 

 "itc. (equivalent to the French il y , etc.), are generally ex- 

 pressed by the auxiliary esserc, before which the adverbs si or 

 vi are placed. Esserc, in this case, must agree in number and 

 gender with the noun that accompanies it ; as 



C' 6 or v' e, there is. 



Ci soiio or vi sono, there are. 



C' era or v' era, there was. 



C' erauo or v' erano, there were. 



Ci fu or vi fu, there teas. 



And so of the other tenses : 



C' e or v' e u-na gran quan-ti-ta, (here 



is a. great quantity. 

 Ci so-no or vi so-no d^l-le per-so-ne, 



there are person*. 

 C' c-ra li-na vol-ta un sa-vio Grfi-co, 



there \cas once a, wise Greek. 

 V t'-ra-no de 1 po-po-li, there tcere 



nations. 



Ci furono or vi f urono, there were. 

 C' e or v' e stato or state, there IMS 



been. 

 Ci or vi sono steti or state, there 



hare been. 



C' 6 sta-te u-na can-te-tri-ce, there 



ho* been a (female) singer. 

 Ci so-no std-ti de' prin-ci-pi, there 



hare been prince*. 

 C' e or v' 6 ec-ci or ev-vi qni un 



qual-che me-di-co ? is there 



here some physician f 



Noto that Sc-ci and &v-vi in the last sentence are written 

 thus according to tho rule that a monosyllable like , in com- 

 positions, loses its accent and doubles the initial (unless an s 

 Impure) of the suffixed word. 



In similar cases avere (with vi before it) may be used for 

 esstre, and even stand in the singular, though the accompany- 

 ing noun is in the plural ; as 



Vha (for v' banno) de' prin-ci-pi, V ha or hav-vi mol-te gen-te po-ve- 

 there are princes. ra, there are many poor people. 



V ha mol-te co-se, there are many I M61-ti sol-da-ti v' a-vrf-a, there were 

 things. many soldiers. 



When there is, there are, and similar phrases, have the 



worda tome, torn* of it, torn* of them, of it, of that, of them, 

 of the number, etc., joined to them, the latter word* most be 

 expreeaed by the particle ne, equivalent to the French en, mod 

 the adverb* a and vi changed into and ve before ne ; ae 



C* M *i-no BxSl-ti. there are MM 



o/tam. 

 Von n' A-ia-no ch do-*, (a** 



WAV only ttro of thtm. 

 M*-dl.ci qoi BOO oe ne a*-BO. Ukrv 



ore net any phyetriiu k*r*. 

 Son orrf-do elM ? B' ib-bU, I e> 



1...I Itl.rit II.:' fUH <.,< ! f.f 



Ce B' 4 or ve a' e, (here it some. 



Ce ne sono or ve ae aoao, tktrt are 



ome. 

 Ce n' era or ve B' era, there o 



MM, 

 Cen' erano orven 1 erano, (* *re 



ome, etc. 

 Non oe n' * piii, (here it 



more of it. 



There it, there are, etc., in onetimes erpreeead bj n di, 



ri dan-no, etc. (from dd-rc, to give) ; ae 



Non si di al n>6n-do o6-a peg-gio-re, there it mot eayOfaf VVTM fa 



(he world. 



Si dan-ne di qtw'l-li ebe so-Bten-gono , there or* torn* who moutbun . 

 Dan-no-si qui de' gran oom-mer-d&n-U P are (Aere yreat merc/uin h*r* t 



The words ci and vi (here, there, in thii or that place) 

 merely being local adverbs, it is dear that they mart be rap- 

 pressed when speaking of time; ae 



P. un mrf-ae, so-no dti-o an-ni, it i a month, it it two yean. 

 I'o-chi m^-si m>-uo (or fa), it ie a few month*, or a /etc month* 090. 

 n Ml pz-7O, cbe non 1' bo ve-dii-to, it it tomt time *inct / heM 



not eeen him. 

 Cib ac-cAd-do dii-e m^-si fa, thu happened two month* ago. 



KEY TO EXEECISES IN LESSONS IN ITALIAN. 

 EXEBCI8E 31. 



1. Voi avete bel tempo per viaggiare. 2. Adeuo abbiamo con- 

 tinuameute belle gioruate. 3. Egli ebbe 1' aano acono un gran giar- 

 dino f nor di citta nel quale trovansi bei fiori e begli alberi da frutti. 

 4. Quel libro tratte della vita di Santo Stefano e di San Giorgio, ed in 

 questo vi sono spiegazioni d' alcuni pami dalle epistole di San Paolo e 

 di San Pietro. 5. Teodosio il Grande mori a Milan o nelle braocia di 

 Sant* Ambrogio. 6. Quello scritto contiene un bel pennero mi 

 vanteggi di cotnraercio. 7. Quei principi sono felici i quali vengono 

 amati de' sudditi. 8. In questo affare bisogna avere gran circospeziooe 

 e gran soraggio. 9. Boma c Cartagine avevano tra di loro gran guerre. 

 10. Demostene era un grand' oratore greco. 11. Egli e un boon 

 giovane, e ha una gran dinposizione d' imparar tutto facilmente. 12. Le 

 gemme sono corpi diafini ; tali sono i diamanti bianchi, il rubino rosso, 

 il zaffiro turchino, lo smeraldo verde, ed il giacinto giallo. 13. Le 

 perle, piccole o grande, crescono in concliiglie, ed i coralii in mare, 

 nella forma di urboscelli. 14. Lo zio mi ha donate un libro franoeae. 



15. Goffredo ha una gran provigiono di vino ungherese ed austriaco. 



16. I cavalli spagnuoli sono cosi cari come 1' inglese. 



LESSONS IN ALGEBRA. XXIX. 



SURDS AND EADICAL QUANTITIES. 



A root whose value cannot be exactly expressed in numbers it 

 called a SURD, or irrational quantity. 



Thus, </2 is a surd, because the square root of 2 cannot be 

 expressed in numbers with perfect exactness. 



In decimals, it is 1*41421356 nearly. 



Every quantity which is not a surd is said to be rational. 



By RADICAL QUANTITIES is meant all quantities which are 

 found under tho radical sign, or which have a fractional index. 



REDUCTION OP EADICAL QUANTITIES. 



CASE I. To reduce a rational quantity to the form of a 

 radical without altering its value. 



liaise the quantity to a power of the game name as the given 

 root, and tlien apply the corresponding radical tign or index. 



EXAMPLE. Reduce a to the form of the nth root. 



The nth power of a is a n . Over this place the radical sign, 

 and it becomes * \/a. 



It is thus reduced to the form of a radical quantity, without 



D 



any alteration of its value. For * A/o n = a" = a. 



N.B. In cases of this kind, where a power is to be reduced to 

 the form of the nth root, it must be raised to the nth power, not 

 of the given letter, but of the poicer of the letter. 



Thus, in the fifth example, Exercise 48, a* is the cube, not of 

 a, but of a 3 . 



CASE II. To reduce quantities which have di/erent indictt 

 o others of the same value having a common index. 



