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 distance 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 trustworthy,—I feel no doubt as to the generally ‘random’
character of the rows of figures displayed by the incommensurable or irrational ratios in question.
As it may interest the reader to see an actual specimen of such a path I append one representing the arrangement of the eight digits from 0 to 7 in the value of π. The data are taken from Mr Shanks' astonishing performance in the calculation of this constant to 707 places of figures (Proc.[ TN: space] of R. S., XXI.[** TN: space] p. 319). Of these, after omitting 8 and 9, there remain 568; the diagram represents the course traced out by following the direction of these as the clue to our path. Many of the steps have of course been taken in opposite directions twice or oftener. The result seems to me to furnish a very fair graphical indication of randomness. I have compared it with corresponding paths furnished by rows of figures taken from logarithmic tables, and in other ways, and find the results to be much the same.
