LOGARITHMS. XXX111 



These rules apply equally well to four or five place tables. In 



of six or seven places the first three litres of the mantissa 



t'ipcd together iii printing, and are therefore more conveniently 



read oil together. Also, it is more convenient to read off six-place 



numbers and iintiloi^s with three :istcad of two in the second 



group ; thus, 7 814 62. 



The difference between a number and a mantissa thus read off, whether 

 audibly or mentally, almost precludes the possibility of mistaking one for the 

 other, BO that lesa strict attention will be required to avoid entering tin- number 

 column of a table with a mantissa, or vice versa. It also avoids the mental or 

 verbal employment of words of instruction. Thus, if a computer reads off 782 

 he knows, nr his assistant usimrthr log tables knows, as soon as the firs' 

 has been read that the logarithm of the number is desired. Conversely, if 

 is off 43857. the first two tiirures alone show that the quantity is a man- 

 tissa, and that the antilou' is required. Tims, no words of instruction need be 



trouizhout an entire computation, and yet no possibility of envi 

 enter. 



I also to be noted that this groupin- -'nt with the symmetrical 



grouping advocated at an earlier pap- ; adapts itself perfectly to the employment 

 of the notation by powers of 10 ; and coincides with the most convenient 

 ing in the five-place tables. 



Powers and Roots by Logarithms. 1 f a = lo" 1 (so that m = 1 

 then (a) n = (io m j n and IOLT <") // ln-4 ". Here both /// and // ma\ 

 be either positive >r negative, and cither an integer or a fraction. 



Hence, to raise any number, whole or fractional, to any power, 

 integral or fractional, and positive or negative, multiply the logarithm 

 of the number by the exponent of the power of the number. 



e a root is a fractional power. i.<. ( ,, ,,,. t he above ride 

 includes ti root. In the case of decimal fractions 



't that the characteristic is negative while the mantissa ; 



itivc. must be regarded. The correct result will In- assured if tin- 



sa and characteristic are separately multiplied (or divided. 



l.y tin- exponent, and the latter result th.-n subtracted from the tor 



;i be further shown. < >pccial procedures will also 



si, when th.- i he power la ft single figure 



ami positive. Thia case takes care of itself without special a 



.iy seem that the use n characteristic eases labor 



it tho foot of page xxviii. Inspeeti..n of 

 . rase in whh -h it does so is where the index of 

 thep<> a single digit. 



