SECT. 17.] Randomness and its scientific treatment. 117 



some of the usual theorems of Probability, where large 

 numbers are concerned, may safely be applied. If it be 

 asked, for instance, whether such a line will ultimately tend 

 to stray indefinitely far from its starting point, Bernoulli s 

 * Law of Large Numbers may be appealed to, in virtue of 

 which we should say that it was excessively unlikely that its 

 divergence should be relatively great. Recur to our gra 

 phical illustration, and consider first the resultant deviation 

 of the point (after a great many steps) right or left of the 

 vertical line through the starting point. Of the eight ad 

 missible motions at each stage two will not affect this relative 

 position, whilst the other six are equally likely to move us a 

 step to the right or to the left. Our resultant drift there 

 fore to the right or left will be analogous to the resultant 

 difference between the number of heads and tails after a 

 great many tosses of a penny. Now the well-known out 

 come of such a number of tosses is that ultimately the 

 proportional approximation to the a priori probability, i.e. to 

 equality of heads and tails, is more and more nearly carried 

 out, but that the absolute deflection is more and more widely 

 displayed. 



Applying this to the case in point, and remembering 

 that the results apply equally to the horizontal and vertical 

 directions, we should say that after any very great number 

 of such steps as those contemplated, the ratio of our dis 

 tance from the starting point to the whole distance travelled 

 will pretty certainly be small, whereas the actual distance 

 from it would be large. We should also say that the longer 

 we continued to produce such a line the more pronounced 

 would these tendencies become. So far as concerns this test, 

 and that afforded by the general appearance of the lines 

 drawn, this last, as above remarked, being tolerably trust 

 worthy, I feel no doubt as to the generally random 



