LESSONS IN LOGARITHMS. 



43 



I.NI>. jv. i. T.n-.-i.i or ttcio, taci, tooe ; taoctimo or tooiamo, Uodte, 

 ticoioiio or taciouo. Imp. Tucd or tacla, taodri, tact 1 v a or taodaj 

 taoeraino, taoev*t, tueevano. Ind. I'ret. Taoqui, Uoditi, tioquo ; 

 UoJmmo, tact'tito. tacquero. Fut. Tacero, tnceral. taceri; taoerdmo, 

 taoarrfto. taceranuo. C'ond. Pr. Tucordl, taoerdsti, Uocrdbbe j taoe- 

 renimo, Uoonfuto, taeere~bbero. 



IMP. Taci, tiocia or tacia; tacoiimo, tact'te, tiooiano or Uciano. 



rr. Che taccia or tiiciu ; chu tacchi, tacia or taci ; die taccia 

 or tacia. Che taooiamo or taciauio ; cbo taciute ; cho tacciouo or 

 taciauo. Imp. Che tacdssi, clio tacdssi, cbe tactae; cho tacduimo, 

 oh* tao&to, che tacdssoro. 



After this example conjugate retacere, to be silent again. 



16. The irregular verb tenure, to hold, is thus conjugated : 



IMF. SimpU Tenset. Pres. Tendre, to hold. Pres. Gerund. Tendndo, 



holding. Put Part. Teuuto, hJd. Compound Tenses. Past. Avdre 



touuto, (o fuit' held. Past Gerund. Avendo touiito, having held. 



/'res. Tdngo, tidni, tidne ; tcuiuwo, tendtc, tdngono. Imp. 

 Tendva or tanda ; tenovi ; tcnovu, tenda or tonfa. Tencvitno ; tcnovato ; 

 tendvano or teni-ano. Jnd. Pt. Tdnni, tendsti, -tdnno; toui'-iuiuo, 

 tendste, tdunero. Put. Terra, terrai, terra ; terrdmo, terrdte, terranno. 

 Coiid. Prts. Terrdi or terria, terrdsti, terrdbbe or terria; trrdmmo, 

 terrosto, terrdbbero or terriano. 



IMP. Tidni, tdngn ; tcniaino, tcniiite, tdngano. 



SUB. Pret. Che ton^a, che tdnga, che touga; che teniamo, che 

 teniato, che tdoguuo. Imp. Che tendssi, che teudssi, che tondssc ; cho 

 tendssimo, che tendste, cho tcndssero. 



After this example conjugate the following irregular verba : 



Appartendre, to belong. 

 Astcndrsi, to abstain. 

 Attendre, to attain. 

 Contendre, to refrain. 

 Detendre. to detain. 

 Intertendre, to d(atn. 

 Mautendre, to maintain. 



Ottenere, to obtain. 

 Pertendre, to belong. 

 Eattendre, to stop. 

 Bitendre, to retain. 

 Soprattendro, to retain. 

 Sostenere, to support. 

 Trattendre, to entertain. 



17. The irregular verb valdre, to bo worth, is thus conju- 

 gated : 



IHT. Simple Tense*. Pres. Vale're, to b worth. Prw. Gerund. 



Valdndo, being tcortJt. Post Part. Valuta, been, worth. Compound 



Tenses. Past, Avdre, valiito, to liave been toorth. Past Gerun<. Avdndo 

 valiito, having been worth. 



IND. Pres. Vdglio or valgo, vdli, vale or val ; vagliamo or valiamo, 

 valdte, vagliono or vulgono. Jmp. Valcva or valda, valdvi, vnldva or 

 valda; valevaino, valevate, valdvano or valdano. Ind. Pret. Vdlsi, 

 valdsti, yalse ; yalemmo, valdsto, vulsero. Put. Varro, varriii, varra ; 

 varrdmo, varrdto, varranno. Cond. Pres. Varrdi, varrdsti, varrdbbe; 

 Tarrdmmo, varrdsto, varrdbbero. 



IMP. Vali, vaglia or vjilga ; vagliamo or valiamo, valdte, vagliano or 

 valgano. 



SUB. Pres. Che vaglia or valga, che vaglia or valga, che vaglia or 

 viilga ; che vagliamo or valiamo, che vagliate or valiate, che vagliano or 

 valiano. Imp. Che valessi, che valdssi, che valdsse ; che valdssimo, che 

 valdste, cho valdasero. 



After this example conjugate the following irregular verba : 



Equivalere, to be equivalent. 

 Prevaldre, to preroil. 



Prevaldrsi, to tafc advantage. 

 Kivaldrsi, to recover. 



VOCABULARY. 



EXKBCISI 45. 



1. Fu domandsto ad nno, percbe avemo poato on* Borda? 

 II quale rinjK>8e : La Bposai tale, oredendo ohe ool tempo doreaao 

 anche divontar mutola. 



2. Un gontiluomo vole'ndo schcrzire oon on actuto barbiere, 

 BUO vicino, gli disae : Quanti goffi iete nella roctra Btrada t Al 

 quale il barbiero riupoHo : No aiamo inci'rcu. ana dozzfna Benza 

 oontaro voosignoria. 



3. Un ricco goffo esae'ndosi fatto efnguir in marmo, motr6 

 qnella figura ad on arai'co BUO, egli domandd ae lo scnltoro 

 avova ben incontrato la rassomjgluinza ? A cui 1' altro rigpooe, 

 Porfettamento corto, perche vi rassomiglia in anima e in oorpo. 



LESSONS IN LOGARITHMS. I. 



DERIVATION OF NAME USB NATURE OF POWERS. 



1. Derivation of the Name. The word "logarithm " is de- 

 rived from two Greek words, signifying number and ratio. The 

 fundamental theory of the system is that a certain fixed num- 

 ber, called a base, raised to the proper power, may be made to 

 represent any number required. 



2. Use of the Method. By the use of logarithms, the more 

 tedious calculations of arithmetic are simplified, the longer 

 processes of multiplication and division being converted into 

 the shorter and easier processes of addition or subtraction, and 

 a simple method provided for the otherwise difficult operations 

 of involution or evolution. 



3. Nature of Powers. If unity be multiplied by any number, 

 the product is called the first power of the number ; thus 



6x1 = 6, the first power. 



If the first power be multiplied by the number, the product ia 

 called the second power, or square ; thus 



6 x 6 = 36, the second power. 



This is also written 6 2 , the figure written above the line being 

 called the index of the power, because it indicates the times 

 which the number has been repeated to form that power. 



If the second power be multiplied again by the original num- 

 ber, the product is called the third power, or cube ; thus 



6 x 6 x 6 = C = 216 ; 



and so on. Hence the following table will show the powers of 

 the number 6 : 



6 xl=^6 1 J = 6, the 1st power. 

 6 l x 6 = 6> = 36, the 2nd power. 

 6 x 6 = 6 = 21C,the 3rd power. 



ff> x 6 = 6* = 1296, the 4th power. 

 6* x 6 = 6 s = 7776, the 5th power. 

 ffi x 6 = 6 =46656, the 6th power. 



This process is called involution. It is obvious that it may 

 be carried to any extent, and that by it ia provided an abbrevi- 

 ated method of writing and dealing with large numbers. Thus, 

 for the fifth power of 6, which is 7776, we write 6 s ; and if we 

 wish to multiply 7776 by 1296, we do so by means of 6* and 

 6 4 , and obtain the result, as wo shall presently prove, in the 

 form 6 9 . 



4. Nature of Roots. We have seen that the products ob- 

 tained by multiplying a number by itself over and over again 

 are called its powers. The number itself, in its relation to these 

 powers, is called the root. Thus, while 36 is the square of 6, 6 

 is called the square root of 36. So, while 216 ia the cube of 

 6, 6 is the cube root of 216. So, again, 1296 is the fourth 

 power of 6, 6 ia the fourth root of 1296 ; and so on to any 

 extent. The process by which the root is obtained from any 

 number is called evolution. Wo may remark that, while in- 

 volution is possible for any number, evolution is only possible 

 for those numbers which are themselves exact potcers of smaller 

 numbers. 



5. We have remarked above that, in indicating the power of 

 a number, a small figure is written above the line. Thus, 6* 

 indicates that four sixes have been multiplied together to form 

 what is called the fourth power of 6. The same method ia 

 employed to indicate evolution, but in this case the indices are 

 fractions whoso numerators are unity, and whose denominators 

 indicate the root which has to be extracted ; thus, while 



6* = 1296 = 4th power of 6; 12%t = 6 = 4th root of 1296. 

 So again 



C* = 777C 5th power of 6 ; 777G* = 6 = 5th root of 7778. 



