SECT. 16.] Randomness and its scientific treatment. 115 



these directions let them be numbered from to 7, and 

 let us say that a line measured due north shall be de 

 signated by the figure 0, north-east by 1, and so on. The 

 selection amongst these numbers, and therefore directions, at 

 &amp;lt;every corner, might be handed over to a die with eight faces ; 

 but for the purpose of the illustration in view we select the 

 digits to 7 as they present themselves in the calculated 

 value of TT. The sort of path along which we should 

 travel by a series of such steps thus taken at random 

 may be readily conceived; it is given at the end of this 

 -chapter. 



For the purpose with which this illustration was pro- 

 pqsed, viz. the graphical display of the succession of digits 

 in any one of the incommensurable constants of arithmetic 

 or geometry, the above may suffice. After actually testing 

 some of them in this way they seem to me, so far as the eye, 

 or the theoretical principles to be presently mentioned, are 

 any guide, to answer quite fairly to the description of ran 

 domness. 



16. As we are on the subject, however, it seems worth 

 going farther by enquiring how near we could get to the 

 ideal of randomness of direction. To carry this out com 

 pletely two improvements must be made. For one thing, 

 instead of confining ourselves to eight directions we must 

 ;admit an infinite number. This would offer no great diffi 

 culty ; for instead of employing a small number of digits we 

 should merely have to use some kind of circular teetotum 

 which would rest indifferently in any direction. But in the 

 next place instead of short finite steps we must suppose them 

 indefinitely short. It is here that the actual unattainability 

 makes itself felt. We are familiar enough with the device, 

 employed by Newton, of passing from the discontinuous 

 polygon to the continuous curve. But we can resort to this 



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