228 



THE POPULAR EDUCATOR. 



(1) To many adjectives formed immediately from verbal stems ; 



as, oj3Aoj37js, uninjured ; avrapKrjs, self -sufficient. 



(2) To adjectives whose second component has arisen from a 



substantive in es (nominative oj); as, Sf/coerT/s, ten-year- 

 old ; KaKorjOris, bad-mannered. 



Without changing its nature a verb cannot be combined with 

 any word except a preposition. If another word is united with 

 a verbal stem, the two unite to form a noun ; thus, out of \idos, 

 a stone, and /3aAAo>, I throw, is formed Ai0o/3oAoj, a stone- 

 throv'cr. Hence a verb may be formed, as \idof3oKfu, I throw 

 stones. So from vavs and fj.axo/J.ai we have vav/j.axos, a sea- 

 fighter, and thence cau^axe&i, I fight by sea; also from fv and 

 fpy come fvfpyerrjs, a benefactor, and tvepytreu, I act as a bene- 

 factor. 



A substantive with an abstract signification may unite with a 

 preposition only by retaining its own termination ; thus /SovAij, 

 a determination, becomes irpo$ov\r\, a pre-ordination ; in every 

 other combination an abstract noun must assume a derivation- 

 ending ; thus \i6os and /3oA?7 (/3oAAco) give rise to Ai0o/3oAio, 

 stone-throwing ; vavs and /J.ax?l give rise to vav/j.axia, a sea- 

 fiyht ; and e and irpats give rise to tuirpafya, a good condition 

 (well-being, weal). 



In regard to signification, compounds may be divided into 

 three classes, determinatives, attributives, and objectives. The 

 determinatives are those compounds in which the secondary 

 component determines the exact meaning of the primary, and 

 in these the second word is the primary or chief word. These 

 compounds are the least numerous; as 6fio5ov\os, a fellow-slave ; 

 cLKpoiroKis, the lofty city (acropolis). 



Attributive Compounds. 



The attributives are those in which also the second word is 

 determined by the first, but the idea formed by the two is 

 attributed as a quality to another word ; thus 6/j.orpottos signifies 

 not the same kind (rpoiros), but being of the same kind, having 

 the same disposition ; and p.axP 3 X fl P * s n t a 1 W} 9 hand, but 

 having a long hand or being long-handed. 



Objective Compounds. 



The objectives are those in which one element is governed by 

 the other, the latter being the object to the former ; thus 5en- 

 Sei/xcov, superstitious, god-fearing, where, as in god-fearing, 8ai/juav 

 is governed by Seui, and the word is equivalent to rovs Saifnovas 

 SaSi&js, fearing the divinities. So rjvioxos, rein-holding, is the 

 same as ra -Sjvia ex wv - ^ n the same manner consider \oyoypa<t>o*, 

 speech-writer (historian or fabulist) ; a|ioAoyor, worthy of record; 

 and x 6 '/* 01 "?! hand-made that is, made by the hand, x f P ffl 

 -iroinros. Sometimes the first component is the object, some- 

 times the second. Especially common are compounds with the 

 prefix av (avev, Latin sine, without), which before consonants 

 becomes a, and which, on account of its negative or privative 

 force, is termed alpha privative ; as, aypa<pos, unwritten; a/xTjroip, 

 motlicrless (or in form more exactly, unmotherly). 



The prefix i>, ivell, and the prefix Svs, hardly, with difficulty, 

 form many compounds : for example, tvroKos, easily-bearing ; 

 8it<rapTTos, displeased. 



VERBAL ADJECTIVES. 



Verbal adjectives have two endings, one in -rot, the other in 

 reos. Those in ros resemble in signification the Latin participle 

 in. -tus, as TroiTjros (foetus), that is, made ; so ypairros (script us), 

 written. Many, and perhaps the greater number of them, more 

 nearly approach the Latin adjectives in -bilis, as flau^acrTos 

 (mirabilis), admirable ; or they express a simple possibility, as 

 oparos, visible, a,n object that may be seen ; UKOVCTTOS, audible. 



Verbal adjectives in TEOJ have the same force as the Latin 

 participles in -dus. and denote duty or necessity, as 

 (dandus), must be given. The adjectives in reos, like the parti- 

 ciple in -dus, has three genders, so as to agree with any noun 

 that may be joined with it ; they may also be used in the neuter 

 in a general way, as signifying necessity ; thus, avi)p Aureos eaTi 

 (Lat. vir solvendus est), the man must be set free ; Tifj.rjrea ffnt/ 

 TJ aptrrj (Lat. virtus Jionoranda est), virtue must be honoured ; 

 ypxirreov TTIV (Lat. scribendum est), it is necessary to write. 



Both these adjectives 'are formed from the verbal stem. An 



easy practical way to form them is to change the termination oi 



the first aorist passive, Deis, into TOJ or reos ; as 



Ayo>, \vOfis. AUTOS, 



TJ/A7/TOJ, 



LESSONS IN LOGARITHMS. IV. 



TABLES OP LOGARITHMS. 



THE following tables will be found very useful, not only to 

 students who are endeavouring to make themselves acquainted 

 with logarithms, but also to persons who are desirous of abridg- 

 ing calculations of any description, especially those connected 

 with the mathematical and philosophical sciences. The first 

 table, called Table of Logarithms, contains the mantissce of the 

 logarithms of all numbers from 1 to 10,000, according to the 

 common system, of which the base is 10. The decimal part of a 

 logarithm is called its mantissa, and the integral part is called 

 its index or characteristic. Thus in the logarithms 0'477121, 

 1-041393, and 3'005609, the decimal parts "477121, -041393, and 

 005609 are the mantissae ; and the integral parts, 0, 1, and 3, 

 are the indices or characteristics. 



The mantissao of the logarithms in the first table extend only 

 to four decimal places ; but these are reckoned sufficient for 

 ordinary purposes. If, however, a greater degree of accuracy 

 be required than can be obtained from this table, recourse must 

 be had to more extensive tables. Let us now proceed to explain 

 our own tables contained in this and the following lesson. 



In the first vertical column of the table are contained the first 

 two figures of any given number, whose logarithm is required, 

 within the range above mentioned. In the next ten vertical 

 columns is contained the third figure of any such number ; 

 these ten columns are headed Third Figure. In the next nine 

 vertical columns is contained the fourth figure of any such 

 number ; and these nine columns are headed Fourth Figure. 



If the logarithm of a number be required which consists of 

 one figure only, as of the nine digits, seek for that figure with a 

 cipher annexed to it in the first column of the table ; and when 

 it is found, then you will find the mantissa of its logarithm in 

 the same horizontal line in the adjoining column on the right, 

 under the figure marked at the top. To this mantissa prefix 

 the index in the manner described in the preceding lessons, and 

 you will have the required logarithm. Example : Required the 

 logarithm of the number 4. Here, looking for 40 in the first 

 column of the table, you find in the same horizontal line, in the 

 adjoining column on the right, and under at the top, the 

 mantisap. -6021 ; to this mantissa prefix 0, which is the index 

 for units, and you have 0'6021 for the logarithm of the number 

 4. If the logarithm of the number 40 were required, the man- 

 tissa would be the same, but the index would be 1 and the 

 logarithm 1*6021. If the logarithm of 400 were required, the 

 mantissa would still be the same ; but the index would be 2, and 

 the logarithm 2'6021 ; and so on. 



If the logarithm of a number be required which consists of 

 two figures only, as of all numbers between 10 and 99, seek for 

 that number in the first column of the table ; and when you 

 have found it, the mantissa of its logarithm you will find in the 

 same horizontal line in the adjoining column on the right, under 

 the figure marked at the top. To this mantissa prefix the 

 index as before, and you will have the complete logarithm. 

 Thus : Required the logarithm of the number 78. Here, looking 

 for 78 in the first column of the table, you find in the same hori- 

 zontal line, in the adjoining column on the right, and under 

 at the top, the mantissa - 8921 ; to this mantissa prefix 1, which 

 is the index for tens, or for a number consisting of two integer 

 figures, and you have T8921 for the logarithm of the number 78. 

 If the logarithm of the number 7 '8 were required, the mantissa 

 would be the same, but the index would be 0, and the logarithm 

 0-S921. If the logarithm of the number '78 were required, the 

 mantissa would still be the same; but the index would be 1, and 

 the logarithm 1-8921; and so on. 



If the logarithm of a number be required which consists of 

 three figures, as of all numbers between 100 and 999, seek for 

 the first two figures of the number as in the preceding case, 

 that is, in the first column of the table ; and when these are 

 found, you will then find the mantissa of its logarithm in the 

 same horizontal line in one of the ten adjoining columns on 

 the right, under the third figure of the number at the top. To 

 this prefix the proper index, and you will have the logarithm 

 required. Thus let the logarithm of 476 be required. Here, 

 looking for 47 in the first column of the table, you find in one 

 of the ten adjoining columns on the right, and under 6 at the 

 top, the mantissa *6776 ; to this prefix 2, which is the index for 

 hundreds, or for a number consisting of three integer figures, 



