the Development of his Theory. 479 



Descriptio (1614) and the Constructio (1619) is to be found 

 in Prop. 1 of the Descriptio : — 



" The logarithmes of proportionall numbers and quantities 

 are equally differing "*. 



And in Section 36 of the Constructio it appears as : — 



" The logarithms of similarly proportioned sines are equi- 

 different." 



Grlaisher has introduced a convenient notation n] r x for 

 Napier's logarithm, in this system, when the radius is 10 r . 

 He also uses Sin r # for the line-sine of the angle x, when the 

 radius is 10 r , and he keeps the symbol sin x for the sine in 

 the modern sense of the term. With this notation we have 



Sin r x 



sin x = 



10 r 



In this paper I follow his notation, and log e # is used in 

 its modern sense for the logarithm of x to the base e, the 

 system commonly called hyperbolic logarithms. 



The fundamental theorem, referred to above, can now be 

 stated as follows : — 



If a : b = c : d, then 



nl r a— nl r 6 = nl r c— nl, -d (1) 



Also we are given that 



nl r 10'=0 (2) 



Napier's Canon consists of a Table of Logarithms in which 

 (1) and (2) are satisfied. His definition of the logarithm 

 by means of the velocities of two points moving in two 

 different lines leads us to the formula 



nl r ^= A — 



,& (t)- 



But, of course, neither this, nor the fact that his function 

 n\ r x has —1 for its differential coefficient, when ^=10 r , 

 could be known in his time. 



The Third Stage. 

 § 4. Since uv : u = v : 1, we have 



nl r (uv) — nl r u = nl r v — nl r 1 . 

 Thus nl r (uv) = nl r u + nil, v — nl r 1, 



and it must be remembered that nl r 1 is not zero. 



When r=7, nl r 1= 161180896'38 (cf. Constructio, Section 

 53). 



Similarly, n\ r (u/v) =nl r w — nl r v + n] r 1. 



* In quoting the Descriptio I follow Wright's version, and for the 

 Constructio I adopt Macdonald's. 



