LOGARITHMS. 



Defcript'tan and Ufe of the preceding Table. — In thi- above 

 table are contained the logarithm of all numbers, from i to 

 10,000, which may be found by infpedlion, according to 

 the method defcribed below ; but it will bo proper, before 

 we enter upon that fubjetl, to make a few remarks with 

 regard to the index, or characleriilic, of logarithms, which 

 are omitted throughout, and mull therefore be fupplied by 

 the operator, according as the cafe may require. It has 

 been fhewn that the bife, or radix of the fyftera, is 10 ; and 

 fince 



10'' = I, 10' = 10, 10' = 100, 10' = 1000, &c. 

 therefore the log. of i =0, the log. of 10 — i, the log. of 

 TOO = 3, the log. of 1000 -1= 3, &c ; and, confequently, 

 the logarithm of any number between f. and 10 has its 

 logarithm greater than o, and lefs than i ; a number be- 

 tween 10 and 100 has its logarithm greater than r, and lefs 

 than 1 ; between lOO and 1000 the logarithm is greater 

 than 2, and lefs than 3, and fo on ; therefore, the integral 

 part of the logarithm, or its index, is always one lefs than 

 the number of its integral places. Again, fince 



I . I _, I _, 



— == 10 ■, = 10 -, — — = 10 ', 



10 100 1000 



it follows, that the logarithm of . I = — i,of.oi = — 2, of 

 .001 = — 3, &c. ; confequently, the logarithm of a number 

 between i and .1 has its index properly o, and its decimal 

 part negative ; but for the greater convenience, and this is one 

 great advantage attending Briggs's logarithms, we may af- 

 fume the index negative, and the decimal part pofitive ; that 

 is, inftead of fubtracting the decimal part from unity, and 

 making the refult negative, we retain the decimal as it 

 arifes, and make the index negative : whence, the logarithm 

 of a decimal greater than . 1 , has its index ::= — i ; if it be 

 lefs than .1, but greater than 01, the index is — 2 ; if it 

 be lefs than .01, but greater than .001, the index is — ■ 3 ; 

 and fo on : whence it follows, that the index of the loga- 

 rithm of any decimal is negative, and always one more than 

 the number of ciphers which precede the firfl efFeftive figure. 

 Or both rules, t/'z. for integers and decimals, may be re- 

 duced to one, which is as follows. The index of the loga- 

 rithm of any number is always equal to the number of 

 places that the decimal point is diftant from the unit's place, 

 being pofitive if the decimal point be to the right of the 

 unit's place, and negative if it be to the left of it. What 

 has been faid will be illuftrated by the following examples : 



Numbers. Logarithms. 



34560 



34560 



34560 



34560 



3.4560 



.34560 



.034560 



.0034560 



.00034560 



4-5385737 



3-5385737 



2-5385737 



1-5385737 



0-5385737 



-'•5385737 



-2-5385737 



-3-5385737 



-4-5385737 



Thefe examples will illuftrate all that has been faid with 

 regard to the index, anJ at the fame time will fhew the 

 great advantage of the prefent fyftem of logarithms ; for 

 here the tabular part of the logarithm is tlie fame through- 

 out, whereas with any other radix, each of the numbers would 

 bave required a different logarithm ; and, conlc-quently, much 

 more extenfive tables tlian any of thofe now in common 

 ufe would be neceffary under thofe circumftances. 



To find the logarithm of any number by the table. — If the 

 number confifts of lefs than three figures, annex a cipher to 

 it, or two if neceffary, confidering it as a dacimal, and look 



for the number thus increafed in one of the firft columns of 

 the table, marked N, and the number in the adjacent column 

 is the decimal part of the logarithm, to which prefix the 

 proper index according to the above rule. 



If the number confills of three figures, it may be found 

 immediately in one of the firll columns, and its logarithm 

 in the adjacent column, to which prefix the proper index a? 

 above. 



If the number confills of four figures, look for the firll 

 three in the column marked N, and feek the fourth figure 

 in the line at the head of the page ; and trace it down t(> 

 the line in which the three firil figures are found, and the 

 meeting of the twa lines will give the logarithm required ; 

 to which prefix the proper index. Thus, 



The log. 34 = log. 34.0 ::x 1.5314789 



The log. 6 = log. 6.00 = 0.7781513 



log. 456 — 2.6589648 



log. 4569 - 3.6590506 



log. 45.69 = 1.6590506. 



If the number confills of more than four places, find the 

 logarithm anfwering to the firil four as above, and for the 

 rclt multiply the number Handing in the correfponding 

 column of difference, by the remaining figures of the pro- 

 pofed number, and cut off from the right hand of the pro- 

 duft as many figures as the multiplier confills of, and add 

 the other part of it to the right-hand figures of the loga- 

 rithm before found ; then prefix to that fum the proper 

 index, according to the rule above given. Thus, to find 

 the logarithm of 34.6782 ; 



'og- 34-67 = 1.5391604 Diff. = J25 



102 82 



^^g- 34-6782 = 1. 5391 706 



250 



1000 



102(50 



and in the fame manner the logarithm of any number what- 

 ever may be found. 



To find the number anftvering to any given logarithm by the 

 table. — Seek for the decimal part of the logarithm in one of 

 the columns of the table, and if it be found there exaSly, 

 the correfponding number is that required, the firll three 

 figures of which will be found in the column marked N, and 

 the fourth in the head line of the table. Then point off 

 the proper number of integers or decimals by the converfe 

 of the rule given in the preceding article, ws. the unit's 

 place mull Hand fo many places to the right or left of the 

 firil figure, as is denoted by the index ; to the right if that 

 index be pofitive, and to the left if negative. 



Thus, the natural number anfwering to the logarithms 



2-5434472 is 349-5 

 -2-5434472 IS 0.03495 



when 9 in the firft, and o. in the fecond, are made the places 

 of units agreeably to the rule. Bui if the logarithm be not 

 found exactly in the table, then feek the next greater and the 

 next lefs, as alfo the difference between the lefs and the given 

 logarithm, and between the lefs and the greater ; which 

 will be found in the correfponding column of difference ; 

 divide the former difference by the latter, and annex the 

 quotient to the right-hand of the four figures before taken 

 out, which will be the number required, remembering to 

 point off the decimals according to the rule. — Note, The 

 above quotient cannot be depended upon for more than two 

 place? 



9 Fino 



