MATH KM \TICS. ALGEBRA. 



[USE OF THE TABLES. 



mast add 1 to the 662, and thus wo obtain for the man- 

 tun of this logarithm -0030221. 



i hare already explained tin- principle of doing this 

 rinciplo, wo hall obtain* 



- -0620182 



- 30U2-.I182 



- 6-0029182 



(}. To find the characteristic. 

 Vie have already <-\ 

 if we apply that pn 



log. 46(117 



; | UQHOO. 



log. -00046017 - 4-0029182 

 log. -40017 - 1 0029182 



If you examine these eases, you will find that they 

 suggest the following rule: "Place your pen between 

 the first and second riatTRR (NOT cipher), and count one 

 for each figure or cipher, until you come to the decimal 

 point; the number this gives will be the characteristic : if 

 you count to the riyht, the characteristic is positive ; if to 

 the left, the characteristic is negative. Thus, in finding 

 log. 4 : t*il7, if you place your pen between the first ) 

 (4), and second (6), it falls on the decimal point ; in this 

 case, therefore, there is no characteristic. Next, in the 

 case of log. 4001 7. place your pen between (4) and (6), 



and count ,' 10 .j ' the characteristic is 3 ; and as you 



count to the right, it is plus 3. Next, in the case log. 

 4001700. here the decimal point falls behind the last 



cipher. Hence, counting as before, we have i 123450 



and the characteristic is phis 6, Again, in the case, 

 log. -00040017, the first fynre is, as before, 4 Hence 



"000-4 601T- 

 counting, we have ^ ' Dut noro we count to 



the left, so that the characteristic is negative or 4. 

 Again, in the case, log. -4601. we have , ' ' and the 



characteristic is 1. 



Instead of writing log. -00046017 * 4-6020182, this is 

 frequently writ '182. To explain this, observe 



that 4-0629182 means 4 + -6629182, which clearly 

 equals 6 + -6620182 10, or 6-0029182 10. It is 

 usual to omit the 10. and write 6 -66201 82 : no ex- 

 perienced calculator would forget the 10, although it 

 is not written down ; but as this tract is intended for be- 

 ginners, we shall never omit the 10, but as it may 

 suit our purpose write log. -00046017 = 4-6620182 or 

 6-0620182 10. 



To find log. 46. Since 46 = 46000 and the table gives 

 mantissa log. 40000 = -6627570. 



.-. log. 46 = 1-6627576. 



Hence, to find the logarithm of any number given in 

 the table, first find the mantissa, and then prefix to it 

 the characteristic, in the manner above explained. 



Find the logarithms of the following numbers from the 

 tables at the end of this chapter, viz. : 



72-643 t-3 8-5072 6315.!! 



84658 ,200 04532 C'5073. 



O57234 >:.s 762 -035872 



35872 ,00 635 (,:;:> 



20000 200 -02 ML':) 



(2). To fin/I the Logarithm of a Number not given 

 in the Tables. 



The rule for the characteristic is the same as given 

 above. For finding the mantissa we proceed as follows : 

 The student will observe that each logarithm on p. 521, 

 differ* from the one before it by 04 or 05. Call this 95, 

 and construct a table of proportional parts as before ex- 

 plained ; this is printed in the column marked 1'P. Wo 

 then proceed as follows : To find log. -0400207. 

 N. L 



6630032 

 I 1 I' 7 (17 



46' -co:; 



BM Art. 17, p. 518. 



N L 



.-. log. -0460267 -"iFOOSOOOO. 

 In practice this is arranged as follows : 



40026 -6530032 

 7 67 



log. -0460267 26630000 



Again, to find lo- 460 3620. 



40 -0630070 

 2 10 

 9 8,6 



log. 400 : 



To find log. 4604-508. 

 40045 

 08 



20631004 



6631825 

 7.6 



log. 460-4508 



In like manner the student may find the logarithms of 

 75-84653 -0027543. 13-OlT.js. 



(3). To find the Number corresponding to a given 



Logarithm. 



It very rarely happens that the logarithm is exactly to 

 be found in the tables. If it be, the only difficulty we 

 have to contend with in such a case is that of fixing the 

 decimal point. For instance, find the number correspond- 

 ing to the logarithm 3'6620080. At the top of the page 

 we have L662 ; this will direct us to the page on which 

 the logarithm will be found ; then, looking in the other 

 part of the table we find 0080, in the column 6 of the 

 line marked 4601 of column N. .'. the number corres- 

 ponding to the mantissa -6620080 is 40016. To fix upon 

 the position of the decimal point, we must modify the 

 rule previously given : place the pen between the first 

 and second figure, and count off as many figures as there 

 are units in the characteristic to the right if the 

 characteristic bo positive, to the left if negative ; and if 

 there are not figures enough, add or prefix as many 



1 l' 1 II '(* 



ciphers as necessary; thus, in the present case, 



1 t> 



.'. the number corresponding to logarithm 3-6620080 is 

 4601.6 ; similarly, that corresponding to 3-6020080 is 

 0046016. 



(4). To find the Number correspondinj to a tjivcn Loyn- 

 rithm, which does not exactly occur in the Tables. We 

 aroceed as follows : 



Find the number corresponding to the logarithm 

 346993191 



The Logarithms 6029277 and 6629372 are in the table ; 

 ,he number, therefore, will be between 46018 and 46010. 

 [t will therefore be the former, with something added 

 on. To fiud this " something " we proceed as follows : 



8628319 



Logarithm next less 46018 6629277 



42 

 Prop, part corresponding to 4 .... 38 



03 .... 3-8 



. . logarithm of 4601843 is tifi2!:il9. 

 .-. logarithm of 460-1843 is 2-6620319. 



(5). To find the Arithmetical complement of the Ltga- 



rithm of a Number. 

 N.B. If x be any number whatever, then the ar. 



. of X = 10 r. 

 Now 10 3-7568274 = 6-2431726. 

 10 2-3907523 = 11-6092171. 

 10 -:i:;jsji:j=. 9-OG7TJ60. 



If you examine thfso, you will find that the snb- 



raction is performed by subtracting the lnxt- ./r/xre (to 



he right hand) from 10, and each of the rest from ; in 



act, to take the first case, wo should proceed as follows : 



from 1 leaves 6, and carry 1 . Then 1 -f- 7 = 8 ; take 



from 10, leaves 2, and carry 1 : but taking R from 10 



s of course the same thing as taking 7 from !), and s.> 



m. The student may, perhaps, think this very obvious, 



put ho will do well not to despise it. 



Hence, to fiud the ar. comp. of tho logarithm of a 



