limited one. The necessity for this restriction may not be obvious at first sight, but the consideration of one or two examples will serve to indicate the point at which it makes itself felt. Suppose that one were asked to choose a number at random, not from a finite range, but from the inexhaustible possibilities of enumeration. In the popular sense of the term,—i.e.[ TN: space] of uttering a number without pausing to choose,—there is no difficulty. But a moment's consideration will show that no arrangement even tending towards ultimately uniform distribution can be secured in this way. No average could be struck with ever increasing steadiness. So with spatial infinity. We can rationally speak of choosing a point at random in a given straight line, area, or volume. But if we suppose the line to have no end, or the selection to be made in infinite space, the basis of ultimate tendency towards what may be called the equally thick deposit of our random points fails us utterly.
Similarly in any other example in which one of the magnitudes is unlimited. Suppose I fling a stick at random in a horizontal plane against a row of iron railings and
inquire for the chance of its passing through without touching 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 velocity below which the stick may pass through without contact, 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 instance; What is the chance that three straight lines, taken or drawn at random, shall be of