LOGARITHMS. 



A r = B ; = 



A'*-' . B' which we have reprefented by M, It will, however, be 



.2.3 ■' proper, before we proceed any farther, to offer a few re- 



= B • whence "'^'''^^ "P°" ^^^^ abfolute value uf this feues, according ta 



' any given radix. Firlt then, iince 



1.2 1.2 1.2.3 ^ 

 each of which gives the fame refult, viz. A x 

 we obtain immediately 



^ ^ B _ (N- i)-x(N- i)' + i(N- i )^ - kc. log. I 4- 

 A 



i «" + : 



: a' + &c. 



(r- i)_i(r- i)^+ i(r~ i) - &c. 

 which is the analytical expreffion for the logarithm of any 

 number N, in fiinftions of itfelf, and the radix of the fyllem ; 

 that is, writing a inltead of N 



1 ^ ^ {a~i)-l(a- lY +i(a-iy- &c. 

 S- (r - I) _ i (,- _ ,)' + i (r - I)' - &c. 



±a -ia' + 



&c. 



ii' 



&c. 



Or,Iog.i±.= ^_^j_,^_^^_^ 



This, however, muft only be confidered as a fimple alge- 

 braical method of expreffing a logarithm ; but it does not 

 always anf.ver the purpofes of calculation, for if a be any 

 number greater than unity, it is obvious that the feries in 

 the numerator will either converge very flowly, or othervvife 

 v.'ill diverge, and the fame with regard to the denommator, 

 fuppofing r to be equal to 10, as it is in the common fyllem ; 

 in facl, the terms of the feries are larger the more remote 

 they are from the beginning ; and confequently no number 

 of them can exhibit, either exaftly or nearly, the true fum. 

 L.et us, therefore, invelligate the method of fubmitting thefe 

 to calculation ; in order to which we will repeat agam our 

 laft feries, viz. 



, + a-ia^ ±^a^- ia-±Sic. 



and here, fmce the denominator is always a conllant quan- 

 tity when the radix of the fyftem is given, we may make 



M = (r - i) - W - ^y + i {>■ - ')'- &c- 

 which renders the above expreffion ftill more fimple, as in 

 that cafe it becomes barely 



log. I + a ■ 



S^ 



+ 



f &c 



•} 



Or, taking a negative, 



log. 



k^i-"- 



M 



Whence again by fubtraftion, 



!«'- &c. X 



icr. 



Now a = 



if, therefore, we fubKitntc in the 



foregoing exprefTicn 



a + I 



inflead of a, it become* 



('• - I) - 5 (»• - 'J' + 5 (»■ - I}' - &c. 

 the denominator and numerator of this fraction are to- 

 tally independent of each other, and therefore r may be 

 aflumed at ple;ifure, and the value of tile whole denomina- 

 tor computed fur any particular magnitude afTigncd to thii 

 letter : or othcrwife, the whole denominator mav be taken 

 equal to any quantity, and the value of /■ itfelf determined 

 by computation. The latter method, at firft fight, appears 

 the moll eligil)li- ; for by affumiug the whole denominator 

 equal to unity, it dilappears entirely, and the expreffion be- 

 comes 



log. ( I + a) = a _ i a' -I- 1 a' - J- a< -f. &c. 



There are, however, inconveniences attending this fyftem, 

 that do not appear upon a flight view of the fubject, but 

 which are notvvithftandii'g very evident upon a farther in- 

 veftigation. In the cafe in which the whole denominator is 

 affumed equal to unity, the value of r, the radix of tliis par- 

 ticular fyllem, is found to be 2,7182818284, &c. and the 



fraftion - becomes =: i. Thefe conllitute what are called 

 M 



hyperbolic logarithms, and which are treated of under that 

 article in the prefent work. We (hall, therefore, enter no 

 farther upon the fubjeft in this place, than is neceffary to 

 Ihew tlie defeft of this fyllem for general purpofes, when 

 compared with that now in common ufc, a deleft which i* 

 by no means compenfated by the trifling advantage at- 

 tending their computation. In the common fyllem the radix 

 r is alTumed equal to 10, the fame as the radix of our fcale 

 of notation ; and hence arifes a moll important advantage, 

 which is, that the logarithm of all numbers expreffed by the 

 fame digits, vi'hether integers, decimals, or mixed of the 

 two, have the fame decimal part ; the only alteration being 

 in the index or charafteriftic of the logarithm. For the 

 radix being 10,0, 1, 3,3, &c. willbelogarithms of 1,10, lO", 

 &c. that is, 10" = I, lo' =: 10, 10 — 100, &c. ; and 

 therefore, to multiply or divide a number by any power of 

 10, we have only to add or lubtraft the number expreffing 

 that power from the integial part of the logarithm, and the 

 decimal part will ftill remain the fame, by which means the 

 tables of lo£rarithms are much more contrafted than they 

 could be with any other radix ; for in the hyperbolic fvffem, 

 or in any other, which has not its radix the fame as that of 

 the fcale of notation, every particular number would require 

 a particular logarithm ; and this circumilance would either 

 fwell the tables to an unmanageable iize, or if they were 

 kept within the prefent limits, frequent computations 

 would become necedary ; fo that in either way it is clear that 

 the advantages of the prefent logarithms much more than 

 counterbalance the extra trouble in computing them. This 



loo 



_ _r ^ S('L——\ -L ' ( ^. I J3 . ! (t H js in fact only confifts in multiplying the hyperbolic logarithm 



M l^'' + 1^ Va +1/ \a + 1/ by a conllant failor ; -vi-z. the reciprocal of the foregoing 



conftant denominator reprefented above by y*» 'he value of 



-f- &c. \ which feries muft neceflarily convergt, becaufe 



the d"nominator of each of the fra£lions is greater than its 

 numerator ; ftill, however, when a is a number of any con- 

 fiderable magnitude, the decreafe in the terms will be fo 

 flow as to render the formula ufelefs for the purppfes of cal- 

 lation. 



At prefent we have affumed the feries which conftitutes 

 the denominator in our firll expreffion a known quantity, 



which, when r = 10, is — v 



= -43429448, 



2.30258^09, &c. 



&c. Hence it is obvious, that different fyftems of logarithm* 

 are connefted together by conftant niultiphers, and by 

 means of which a logarithm may always be converted ."tom 

 one fcale to atiolher. Thus the liyperbohc logarithm of a 

 1.. 1 2 number 



