I cannot but think that there is a similar fallacy in De Morgan's admirably suggestive paper on _Infinity_ (_Camb. Phil. Trans._ Vol. 11.) when he is discussing the "three-point problem":--i.e. given three points taken at random find the chance that they shall form an acute-angled triangle. All that he shows is, that if we start with _one side as given_ and consider the subsequent possible positions of the opposite vertex, there are infinitely as many such positions which would form an acute-angled triangle as an obtuse: but, as before, this is solving a different problem. 9. The nearest approach I can make towards true indefinite randomness, or random selection from true indefiniteness, is as follows. Suppose a circle with a tangent line extended indefinitely in each direction. Now from the centre draw radii at random; in other words, let the semicircumference which lies towards the tangent be ultimately uniformly intersected by the radii. Let these radii be then produced so as to intersect the tangent line, and consider the distribution of these points of intersection. We shall obtain in the result _one_ characteristic of our random distribution; i.e. no portion of this tangent, however small or however remote, but will find itself in the position ultimately of any small portion of the pavement in our supposed continual rainfall. That is, any such elementary patch will become more and more closely dotted over with the points of intersection. But the other essential characteristic, viz. that of ultimately _uniform_ distribution, will be missing. There will be a special form of distribution,--what in fact will have to be discussed in a future chapter under the designation of a 'law of error',--by virtue of which the concentration will tend to be greatest at a certain point (that of contact with the circle), and will thin out from here in each direction according to an easily calculated formula. The existence of such a state of things as this is quite opposed to the conception of true randomness. 10. III. Apart from definitions and what comes of them, perhaps the most important question connected with the conception of Randomness is this: How in any given case are we to determine whether an observed arrangement is to be considered a random one or not? This question will have to be more fully discussed in a future chapter, but we are already in a position to see our way through some of the difficulties involved in it.