156 SCIENCE PROGRESS 



the ingenious manner in which he conceived his related arithme- 

 tical and geometrical series to be developed. 



This latter relation is treated in a manner which strongly 

 recalls Newton's subsequent development of the fluxionary 

 calculus and may fitly be described here, leaving the question of 

 the construction of the tables to be considered later. Trans- 

 lated into modern language and notation, Napier's treatment of 

 the problem proceeds thus : 



C< x >F D 



Imagine two lines A B and C D, A B of length equal to the 

 radius, CD of indefinite length. Let two points start simultane- 

 ously from A and Cwith the same initial velocity. But whilst 

 the velocity of C remains uniform, let that of A decrease in 

 such a way that at any stage of the journey, such as E, its 

 velocity is proportional to the distance E B yet to be described ; 

 when one point has reached F let the other be at E. Then C F 

 is called the logarithm of E B. 



The length E B is taken as the sine of a given arc and A B 

 as the whole radius. It is clear, therefore, that the logarithm of 

 radius — that is, the logarithm of the sine of 90 — is zero and that 

 the logarithms increase as the sines decrease. The connexion 

 between Napierian logarithms and logarithms to the base e — 

 often wrongly called Napierian logarithms — may thus be 

 exhibited in modern notation : 



By definition, 



CF = \ogN EB, 

 i.e. x = \og/vf. 



Also, the velocity of E = d ,™' = y, by hypothesis, since 



Napier takes the constant of proportionality as unity. 

 Hence, integrating 



- log, y = t + k. 



To determine the integration constant we note that when 

 t — o, y = a and therefore 



k = - log, a. 



Hence 



a 

 t = log*- 



