400 Mr. E. A. N. Pochin : 



at an identical result. This expression is usually denoted 

 by e, and accordingly (1 + 1/m) = *V» (1 + l/m) 2 = <? 2 / m , &c. 

 The previous list may therefore be written as follows : — 



The radius vector is When the angle is 



1 



e l/m y m 



e 2/m 2/ 



g3/m 



\ or e 



m 



3/ m 



m/m, or 1. 



We have thus secured a most important quantity e, 

 which affords a simple index relation between the radius 

 vector and its angle. 



Hence we can, as before, multiply together any two values, 

 such as e a l m and e b/m , by adding the angles a/m, b/m, and 

 reading off the value of the radius vector e a/rn+b/v \ which is 

 both appropriate to the angle a/m + b/m, and at the same 

 time, the obvious value of the product. And since this 

 analysis is applicable throughout the spiral, we have shown 

 both experimentally and analytically that 



log A + log B = log AB. 



Perhaps we may with advantage pause one moment to 

 consider the full meaning of e. It is usually defined as the 



sum o£l + l+-^--f|o ? an d though undoubtedly correct, 



I would suggest that this is not really e, but only a method 

 of calculating e. There are, for example, many ways of 

 evaluating ir ; but surely our conception of it must always 

 be the visible circumference over the diameter, and not an 

 infinite series. In the same way the student might, I think, 

 get a clearer view of e by regarding it as the result of unity 

 growing, in a special manner, through uuit angle : as the 

 amount of £1 at compound interest after one year, interest 

 being paid continuously at the rate of one millionth of the 

 capital per one millionth of the time, or even as the first 

 great milestone along the 45° spiral, with " 2*7 miles from 

 London " painted on it in big letters. 



Change of Base. 



Possibly you have never seen a modulus. Here is one 

 made of cardboard. It is called " the modulus of the common 



