XV. Of the construction of logarithmic tables

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 + x ) = ′A x + ″A x 2 + ‴A x 3 + ‴′A x 4 + &c., whence L (1 + y ) = ′A y + ″A y 2 + ‴A y 3 + ‴′A y 4 + &c., and L {(1+ x ) (1+ y )} = L(1+ x + y + xy ), or putting 1+ x = π , =L{1+( x + πy )=′A( x + πy )+″A( x + πy ) 2 +‴A( x + π y) 3 +&c.