C *»7 3 
XV. Of the construction of Logarithmic Tables. By Thomas 
Knight, Esq. Communicated by Taylor Combe, Esq. Sec. 
R. S. 
Read February 27, 1817. 
1. I have endeavoured, in this short Paper, to give a simple 
and connected theory of the construction of logarithms, which 
I think has not hitherto been done. 
Prop. I. 
To find the Logarithm of 1 x.* 
It is not difficult to see that we may assume 
L (1 -|- ce) — 'Ax -j- "Ax* -f "'Ax 3 -|- Ax * &c., whence 
L (1 + f) = 'Ay + "Ay* + '"Ay 3 -f ""Ay* -f &c., and 
L{(i+i)(i +>) } = L( x +x+y+xy ) , or putting 1 +x=v, 
=L{i+C*+xry)}=='A(*+xO')+' , A(®+*5’)‘+'''A(®+ ! r y )'+&c. 
If we substitute these three expansions in the equation 
L(i +x) +L(i -f y)= L{(i +o?) (1 +y)} 
which expresses the nature of logarithms, and compare the 
coefficients of the first power of y, we find 
'A = 'AtT + 2"A7 tx -j- 3'" Anx* + Attx 3 -j- &c. 
or ''A = 'A-f 2 " Ax -j- 3"' Ax* -j- 4""A.r 3 + &c. 
+ 'A| +2"A| + s"'A| . 
whence, by comparing the coefficients of the powers of x, 
'A— 'A, 2"A+ / A=o, 3 " / A-fi2"A=o, ^'"A-^^A—o, &c.; or 
'A='A, "A=— — , '''A=-——-, " >, A=- 3 ~=--, & c. 
2 3 3 4 4 ’ 
* I find that the method of expansion made use of in this Proposition had been 
previously employed by Mr. Spence. 
Ff 2 
