SECT. 7.] Randomness and its scientific treatment. 103 



inquire for the chance of its passing through without touch 

 ing them. The problem bears some analogy to that of the 

 chessmen, and so far as the motion of translation of the 

 stick is concerned (if we begin with this) it presents no 

 difficulty. But as regards the rotation it is otherwise. For 

 any assigned linear velocity there is a certain angular ve 

 locity below which the stick may pass through without con 

 tact, but above which it cannot. And inasmuch as the former 

 range is limited and the latter is unlimited, we encounter 

 the same impossibility as before in endeavouring to conceive 

 a uniform distribution. Of course we might evade this 

 particular difficulty by beginning with an estimate of the 

 angular velocity, when we should have to repeat what has 

 just been said, mutatis mutandis, in reference to the linear 

 velocity. 



7. I am of course aware that there are a variety of 

 problems current which seem to conflict with what has just 

 been said, but they will all submit to explanation. For in 

 stance ; What is the chance that three straight lines, taken 

 or drawn at random, shall be of such lengths as will admit of 

 their forming a triangle ? There are two ways in which we 

 may regard the problem. We may, for one thing, start with 

 the assumption of three lines not greater than a certain 

 length n, and then determine towards what limit the chance 

 tends as n increases unceasingly. Or, we may maintain that 

 the question is merely one of relative proportion of the three 

 lines. We may then start with any magnitude we please to 

 represent one of the lines (for simplicity, say, the longest of 

 them), and consider that all possible shapes of a triangle 

 will be represented by varying the lengths of the other two. 

 In either case we get a definite result without need to make 

 an attempt to conceive any random selection from the in 

 finity of possible length. 



