napier's logarithms : the development of his theory. 141 



point of the theory; and that, if the logarithm of unity is 

 zero, the logarithms of the product and the quotient of two 

 numbers are, respectively, the sum and difference of their 

 separate logarithms. 



§ 9. The Differential Equation satisfied by the logarithm 

 of x. 



We have seen that the theory of the different systems of 

 logarithms described in the previous pages rests upon the 

 fundamental property: — 



If a : b = c : d, then A(a) - A(b) = A( c ) - A(d), 

 where h(x) stands for the logarithm of x. 



The function k(x), therefore, satisfies the equation 



X(x + h) - Hx) = . \(l + ^) -* A(l). 



/t x j h 



v x 



Proceeding to the limit h — >0, of course keeping x fixed, 

 we have y(x) = A where A = y(1)> 



X 



Therefore A(x) = A log e x + B, 



and the system is made definite by adding two other con- 

 ditions. 



In Napier's Canon, writing p for the radius, we have 

 nl x = A loge x + B, 

 with nip = 0, and til' p = — 1. 



Therefore nix = p log e (— 



In Briggs' modification of the system, we have 

 bloc = A loge oc + B, 



with bl p = and bl (-^-) = 10 10 . 

 Thus bl x = 1(P log -^ = 10- log, (^ 



