SECT. 15.] Randomness and its scientific treatment. 113 



whole inches ? The &quot; piece over &quot; which we are measuring 

 may in fact be regarded as an entirely new piece, which had 

 fallen into our hands after that of 31 inches had been 

 measured and done with ; and similarly with every successive 

 piece over, as we proceed to the ever finer and finer divisions. 



Similar remarks may be made about most other incom 

 mensurable quantities, such as irreducible roots. Conceive 

 two straight lines at right angles, and that we lay off a 

 certain number of inches along each of these from the point 

 of intersection; say two and five inches, and join the ex 

 tremities of these so as to form the diagonal of a right-angled 

 triangle. If we proceed to measure this diagonal in terms 

 of either of the other lines we are to all intents and purposes 

 extracting a square root. We should expect, rather than 

 otherwise, to find here, as in the case of TT, that incommen 

 surability and resultant randomness of order in the digits 

 was the rule, and commensurability was the exception. Now 

 and then, as when the two sides were three and four, we 

 should find the diagonal commensurable with them ; but 

 these would be the occasional exceptions, or rather they 

 would be the comparatively finite exceptions amidst the 

 indefinitely numerous cases which furnished the rule. 



15. The best way perhaps of illustrating the truly 

 random character of such a row of figures is by appealing to 

 graphical aid. It is not easy here, any more than in ordinary 

 statistics, to grasp the import of mere figures ; whereas the 

 arrangement of groups of points or lines is much more 

 readily seized. The eye is very quick in detecting any 

 symptoms of regularity in the arrangement, or any tendency 

 to denser aggregation in one direction than in another. 

 How then are we to dispose our figures so as to force them 

 to display their true character ? I should suggest that we 

 set about drawing a line at random; and, since we cannot 

 v. 8 



