402 



Mr. E. A. N. Pochin 



This concludes the graphic treatment of logarithms. Some 

 portions have been made very elementary, and must neces- 

 sarily appear prolix. On the other hand, I have entirely 



/ 1 \ m 

 omitted such questions as the limiting value offlH — J , 



negative characteristics, and other points which are fully 

 explained in the ordinary books. The short descriptions o£ 

 the watch calculator and slide-rule, though not essential, have 

 been deliberately introduced, in the belief that such examples 

 are beneficial. 



I will terminate by mentioning, very briefly, two other 

 properties of the spiral which admit of illustration. 



Differentiation. 



On the left of fig. 4 is the e spiral; the remainder of the 

 construction being self-evident. 



The angle POP' is the increment of the natural log, cor- 

 responding to the increment NP' of the radius vector. 

 Therefore 



^K- limit of ™g?-i. 

 dr NP' r 



On the right of fig. 4 is the " 10 " spiral, drawn, as we 



Fkr. 4. 



0^/A 



have seen, with the roller at an angle whose tangent is 

 1/2-302 .... Therefore in this case 



d log! or 1 M 



dr 



2-3.02. r 



