independent coordinates of assumed continuous and equal facility, so we might conceive our making the attempt to analyse the man's movements into a certain number of independent constituents. We might suppose all the various directions from his starting point, along the ground, to be equally likely; and that when he reaches the shelves the random motion of his hand is to be regulated after the fashion of a shot discharged at random.
The above would be one way of setting about the statement of the problem. But the reader will understand that all which I am here proposing to maintain is that in these, as in every similar case, we always encounter, under this conception of ‘randomness’, at some stage or other, this postulate of ultimate uniformity of distribution over some assigned magnitude: either time; or space, linear, superficial, or solid. But the selection of the stage at which this is to be applied may give rise to considerable difficulty, and even arbitrariness of choice.
§ 5. Some years ago there was a very interesting discussion upon this subject carried on in the mathematical part of the Educational Times (see, especially, Vol. VII.). As not unfrequently happens in mathematics there was an almost
entire accord amongst the various writers as to the assumptions practically to be made in any particular case, and therefore as to the conclusion to be drawn, combined with a very considerable amount of difference as to the axioms and definitions to be employed. Thus Mr M. W. Crofton, with the substantial agreement of Mr Woolhouse, laid it down unhesitatingly that “at random” has “a very clear and definite meaning; one which cannot be better conveyed than by Mr Wilson's definition, ‘according to no law’; and in this sense alone I mean to use it.” According to any scientific interpretation of ‘law’ I should have said that where there was no law there could be no inference. But ultimate tendency towards equality of distribution is as much taken for granted by Mr Crofton as by any one else: in fact he makes this a deduction from his definition:—“As this infinite system of parallels are drawn according to no law, they are as thickly disposed along any part of the [common] perpendicular as along any other” (VII.[ TN: space] p. 85). Mr Crofton holds that any kind of unequal distribution would imply law,—“If the points [on a plane] tended to become denser in any part of the plane than in another, there must be some law attracting them there” (ib.[ TN: space] p. 84). The same view is enforced in his paper on Local Probability (in the Phil.[ TN: space] Trans., Vol. 158). Surely if they tend to become equally dense this is just as much a case of regularity or law.
It may be remarked that wherever any serious practical consequences turn upon duly securing the desired randomness, it is always so contrived that no design or awkwardness or unconscious one-sidedness shall disturb the result. The principal case in point here is of course afforded by games of chance. What we want, when we toss a die, is to secure that all numbers from 1 to 6 shall be equally often represented in the long run, but that no person shall be able to predict the
individual occurrence. We might, in our statement of a problem, as easily postulate ‘a number thought of at random’ as ‘a shot fired at random’, but no one would risk his chances of gain and loss on the supposition that this would be done with continued fairness. Accordingly, we construct a die whose sides are accurately alike, and it is found that we may do almost what we like with this, at any previous stage to that of its issue from the dice box on to the table, without interfering with the random nature of the result.
§ 6. II. Another characteristic in which the scientific conception seems to me to depart from the popular or original signification is the following. The area of distribution which we take into account must be a finite or