LKFlMTInNS AM KX I'LA N ATIONB. xh 



tion of four significant figures throughout will insure entirely sufficient freedom 



from computation error in every case when the number of rejections is less than 



1 total of about 20, and fewer places would not be warranted. 



Similarly five places will suffice : to o.i per better than I per 



it not much better than o.i per cent; six places for work of o.oi per 



cent, and so on. 



As to relaxation of in special cases, it is evident that but little can 



be safely done. The maximum error with n = 7, r = 4, will be 0.35 per cent, 



and an error of $ of this, say, of o.i per cent, will not be of very uncommon 



with only two rejections the error might be o.i per cent. 



Hence, four places cannot In- e< is safe to o.i per cent, even for short 



coinput.i Four places might properly enough be used in short computa- 



tions up to A i xi :uents of course hold for the other rules. 



When logarithms are used for multiplication, division, etc., tabta should be 

 used giving the mantissa to the same number of places of figures as required by 

 the foregoin r direct multiplication and division. Hence, one 



four-place tables for i per cent work, five-place tables for o.i percent 

 work, and so on. The numbers, antilogs, and mantissa should all be carried 

 out to the same number of places. This conforms to the customary and only 

 convenient pra< : 



-ed primarily on the fact, next to be shown, that under them 

 the maximum computation error in the use of logarithms arises chietly from the 

 numbers and antilogs themselves and not sensibly from tin- 

 rejected places in the logarithms. In logarithms, a change of the rth figure by i 

 produces the same fractional error in the antilo.i; whatever its value. \ i/.. 2.4/10', 

 as may be seen easily by inspection of table-. M.-n<-,-. as the maximum r. 

 error in the tabular value of a logarithm is 5 in the (r -f i) place of the man- 

 tissa, which may be doubled by the process of iuterpolatinir. the maximum 

 : Tor in a result due to the maximum rejection error in any mantissa 

 is 2.4/10". Kut if only r places an- retained in the number or antilog, the 

 rnr in it due to rejection <>f its further places is S/lo 1 ', compared 

 to which 2.4/10' Me. Henee, as the mini! :u num- 



1 antilog together is usually about the same as from mantissa, the a< 

 lated error will be due almost wholly to the r. :-'in the numb. 



11 ordinarily be about as many if the 



computation is earned out by logarithms as if by direct multiplication or 

 ove rule is thus justified. 



In addition or subtraction the maximum rejection error will l>e obviously 

 'he (r-f i stated rule that the weakest quan tit v 



shall be carried to four significant figures (or two nne- 



uiallest value of ib- d. . i.ui..n measure or un--. rtamr. 

 result will be 10 in the rth place, which is 100 in the (r + i) place. Hence, 



5n^ 100. .. n = 20. 



The maximum accumulated error would then attain the size of the smallest 

 n me.iM worst possible case would occur, on num- 



ber of i was as great as twenty. Hence, the rule of four places for i 



per cent work, five for o.i per cent work, and so on, as before given, is sufficient. 



