LOGARITHMS. 



Bumbcr is transFormed to the common logarithm, by multi- itig them from one fcale to another ; we will now add on'' 



plying the former by .4342944 ; and the latter is converted example by way of illullration. Let it tlu-refore be 



into the former by multiplying it by 2.30258509. propofed to find the common logarithm of 3. In this caie 



Having faid thus much with regard to advantages of dif- our 

 ferentfyltems of logarithms, and the method of transform- 



M 



2 

 becomes log. 3 = - x 



the computation of which will ftand thus : 

 I 



12 "^3.2' 



I 



I 

 7. 2' 



9. 2' 



11.2" 

 I 



13 • 3'^ 

 I 



35.2'> 

 I 



17.2" 



1. Log. a = 



2. Log. a = 



3. Log. a = 

 4 



Log. - = 

 5. Log. ^ = 



I 



M 

 I 



M 

 2 



M 

 1 



M 

 I 



M ' 

 a 



.0416666666 

 .00625 

 .0011160714 

 .000217013S 

 .0000443892 

 .0000093900 

 .0000020345 

 .0000004487 

 : {{a~ I) - 



+ & 



-} 



ai.2" 

 I 



whence 



ii 



.0000001002 



.0000000227 



.0000000050 

 •549306H22 



(a - ly + i {a - jy - 



{{:-^)+^(:-f^> 



{(^')-'C-t-') 



7. Log. a = log. (a - I) + j^ X {J + 7^ 



8. Log. a = log. (a - j) + ^ X 1^3-^- 



^ZSSJ^' °' ^^'♦'^'^^^ ^ 549306142^ = 

 .4771212, which is the logarithm of 3 required. 



This feries, we have already obfcrved, will only anfwerfor 

 the computation of the logarithms of fmall numbers, in 

 other cafes different feries mull be employed according to 

 tiie particular number under confideration. The limits of this 

 article will not admit of an inveftigation of the feparate cafes. 

 But for the fake of reference it will be uleful to fubjoin a 

 few of the mod ufeful formulas, for which purpofe we avail 

 ourfelves of the fcleftion made by Mr. Bonnycallle, in his 

 valuable treatife of Trigonometry. 



^°2T = M 



+ 

 + 



+ 

 + 



&c 



C-f^)' - - } 



J 



2 (a - I)' 



9. Log. 



log. (a - 2) 4- 



2 

 M 





+ 



+ ■ 



3 (a- i;^ 

 I 



1 3(«-i)'^5(«-i)' 

 To the abore may be added the following, which will be found ufeful on many occafions. 



10. Log. 1 = ^ X |(« - a~') - i {a - fl-') + i (a- a ') - &C.J 



+ &c 



&c.| 





+ &c. 



+ &c, 



} 



11. Log. (a + 2) = log. a .|- - X { J- 



12. Log. (a-^) =log.a- ijx{^+ xy+ i^ + 1^; + &c.j 

 ,3. Log. (.+ .)= log. a ± ^ X {(^-^) + .1 (^y + i (^y -r &c. } 



J4. Log. it = 





/« 



I) -4(:;^'a- ir+iCv/^- 0'- 



&c. 



} 



Thef* 



