116 Randomness and its scientific treatment* [CHAP, v 



device because the ideal, viz. the curve, is as easily drawn 

 (and, I should say, as easily conceived or pictured) as any o1 

 the steps which lead us towards it. But in the case before 

 us it is otherwise. The line in question will remain dis 

 continuous, or rather angular, to the last : for its angles do- 

 not tend even to lose their sharpness, though the fragments 

 which compose them increase in number and diminish in 

 magnitude without any limit. And such an ideal is not con 

 ceivable as an ideal. It is as if we had a rough body under 

 the microscope, and found that as we subjected it to higher 

 and higher powers there was no tendency for the angles to 

 round themselves off. Our random line must remain as 

 spiky as ever, though the size of its spikes of course 

 diminishes without any limit. 



The case therefore seems to be this. It is easy, in words 

 to indicate the conception by speaking of a line which at 

 every instant is as likely to take one direction as another 

 It is easy moreover to draw such a line with any degree 

 of minuteness which we choose to demand. But it is nol 

 possible to conceive or picture the line in its ultimate form l 

 There is in fact no limit here, intelligible to the under 

 standing or picturable by the imagination (corresponding to 

 the asymptote of a curve, or the continuous curve to th&amp;lt; 

 incessantly developing polygon), towards which we find our 

 selves continually approaching, and which therefore we are 

 apt to conceive ourselves as ultimately attaining The usua 

 assumption therefore which underlies the Newtonian in 

 finitesimal geometry and the Differential Calculus, ceases to 

 apply here. 



17. If we like to consider such a line in one of its 

 approximate stages, as above indicated, it seems to me that 



1 Any more than we picture the shape of an equiangular spiral at tht 

 centre. 



