§ 6. And now to proceed to the actual results of the census, and to the calculations based thereon. [I omit Gurney's calculation for auditory cases. Turning to visual ones, he says: - Ed.] The reductio ad absurdum becomes far more striking when we apply the doctrine of chances to visual cases. Out of the 5705 persons taken at random, of whom the above question was asked, only 21 could recall having, in the conditions named and within the specified 12 years, experienced a visual hallucination representing a living person known to them. But two of the 21 had had 2 experiences of the sort; so let us take the total as 23.l That is, the experience has fallen to the lot of one 248th of the group of persons asked, or, if that group be fairly representative, to 1 person in every 248 of the population.3

Let us now see what the proportion of the population who have had such an experience ought to be, on the hypothesis that the similar impressions of recognised voices presented in this book as telepathic were really chance-coincidences. As before in the case of dreams (pp. 210-13), I take cases where the coincidence of the hallucination was with deaths - the reasons for this selection being (1) that death is the prominent event in our telepathic cases; and (2) that for the purpose of an accurate numerical estimate it is important to select an event of a very definite and unmistakable kind, such as only happens once to each individual. Again also, in accordance with the official returns which give 22/1000 asthe annual death-rate, the proportion of anyone's relatives and acquaintances who die in the course of 12 years is taken as 264/1000; and as we have seen (pp. 211-12), it will make no appreciable difference to the calculation whether a person's circle of relatives and acquaintances, any one of whom his hallucination may represent, is large or small.

The probability, then, that a person hallucinated in the way supposed will, by accident, have his hallucination within 12 hours on either side of the death of the relative or acquaintance whom it represents, is 1 in 12x365x1000/264, or 1/165.91. That is to say, each coincidental hallucination of the sort in question implies 16,590 purely subjective cases of the same type. Now our collection includes 21 first-hand1 and well-attested coincidental cases of this kind, which have occurred in this country within the specified time.2 On the hypothesis, therefore, that these cases were accidental, the circle of persons from whom they are drawn ought to supply altogether, in the specified 12 years, 348,390 examples.

1 This is a liberal allowance; for it includes several cases where there was such an amount of anxiety or expectancy on the part of the hallucinated person as would prevent us, if it were present in a coincidental case, from including such a case in our telepathic evidence. In seven of the cases, the form seen was an "after-image" of what had been, for some time previously, part of the perceiver's daily visual experience.

2 It will be Been that one in 248, though a small proportion, is yet quite large enough to make it likely that most of us should casually have heard of a case or two of the kind. For there are probably more than 248 persons whom we are each of us sufficiently near to make it natural that an unusual experience - such as a distinct "spectral illusion" - befalling one of them, should directly or indirectly reach our ears. This is worth noting, because one sometimes hears the statement, "Why I heard the other day of a person who had been disturbed by seeing an apparition of a friend, and nothing came of it," made as though it amounted to a proof that such experiences were common enough to afford scope for any number of marked coincidences.

The next point to decide is the size of the circle from which our coincidental cases are drawn. The number here is not one that it is possible to estimate accurately: what must be done, therefore, is to make sure that our margin is on the side adverse to the telepathic argument, i.e., to take a number clearly in excess of the true one. Our chief means of obtaining information, apart from information derived from our own circle of friends, has been by occcasional requests in newspapers. A million-and-a-half would probably be an outside estimate of the circulation of the papers which have contained our appeals; but it by no means follows that every paragraph in a paper is studied by every person, or by a tenth of the persons, whom the paper reaches. However, I will make the extreme assumption that as many as a quarter of a million of people have by this means become aware of the kind of evidence that was being sought - an assumption which probably arrogates to us who sought it many times as much fame as we really possess; and I will allow another 50,000 for those who have become aware of the object of our work through private channels.

This would raise the number of the circle from whom our evidence is drawn to 300,000, or about 1/80 of the adult population.3 No one, I think, will maintain on reflection, that I am taking too low an estimate. Would anyone, for instance, suppose that if he canvassed the first 1000 adults whom he met in the streets of any large town, he would find that 12 or 13 of them had, within the last three years, been aware of what we wanted, and of the address to which information might be sent? and for rural districts such a supposition would be even more violent. But I am further supposing that this area of 300,000 persons has been drained dry - again an extravagant concession; for though it is easily assumed that anyone who has ever had a "psychical" experience is desirous to publish it abroad, as a matter of fact people do not usually take the trouble to write a letter about family and personal matters to perfect strangers, on the ground of a newspaper appeal; and I have already mentioned that we ourselves know of much evidence which the reluctance or indifference of the parties concerned has made unavailable for our collection; we cannot, therefore, doubt that much more remains unelicited even among those whom our appeal has reached.

A further strong argument for the existence of these unelicited facts is the very large proportion of our actual cases that has been drawn from a circle of our own, unconnected with "psychical" inquiry - from the friends, or the friends' friends, of a group of some half-dozen persons who have had no such experiences themselves, and who have no reason to suppose their friends or their friends' friends better supplied with them than anybody else's.

1 In three of the cases the evidence is not first-hand from the percipient, but is of the nature described in p. 119.

2 Nos. 26, 27, 28, 29, 172, 173, 175, 182, 184, 195, 202, 236, 237, 238, 249, 293, 300, 350, 695, 697, and the case described in p. 107, note. [This is the list as finally decided on by Mr. Gurney, for the purposes of the above calculation, after discussion with Professor Peirce (see papers in Proceedings of American S.P.R. already referred to) I have altered the calculations in the text in accordance with this decision. All these cases will be found in the present volume. - Ed].

3 In the "adult population" I mean to include all persons above fifteen years of age. In the Supplement to the 45th Annual Report of the Registrar-General, p. xix., the proportion of such persons is given as .64 of the whole: which would make their number about 24,000,000.

Here, then, is the conclusion to which we shall be driven, if our coincidental cases were really purely subjective hallucinations, and the coincidence was an accident: - that a circle of 300,000 persons ought to supply altogether in the specified 12 years 348,390 examples of the type in question. That is to say, it ought to have happened on an average to everybody once, and to some people twice, within the given time, distinctly to see an absent relation or acquaintance in a part of space that was actually vacant. But the census has shown that, within the given time, only about 1 in every 248 persons has had such an experience even once. Thus the group of visual coincidental cases now in question, if ascribed to accident, would require either that the subjective hallucinations should be more than 288 times as numerous as they actually are; or else that the circle from whom our coincidental cases are drawn should amount to more than 288 times the assumed size - in other words, that our existence and objects should have been prominently before the minds of every adult member of a population 3½ times as large as the existing one.

Another form of the estimate is as follows. The probability that a person, taken at random, will, in the course of 12 years, have the form of hallucination in question is 1/248; the probability that any assigned member of the general population, and therefore any particular person whose phantasm is seen, will die within 12 hours of an assigned point of time is 22/1000 x 1/365; hence the probability that in the course of 12 years, a hallucination of this form and the death of the person whose phantasm is seen will fall within 12 hours of one another is 1/248 x 22/1000 x 1/365, or 1 in 4,114,545. And the circle from which our coincidental cases are drawn is assumed to be 300,000. From these data it may be calculated that the odds against the occurrence by accident, of as many coincidences of the type in question as the 21 which that circle produced, are about forty million billion trillions to 1. Or, to put it yet another way - the theory of chances, which gives 1 as the most probable number of coincidences of the type in question for every 4,114,545 of the population to yield, will give 6 as the most probable number for the whole adult population to yield, within the given period.

Yet we draw 3½ times that number from a fraction of the adult population which can only by an extravagantly liberal estimate be assumed to amount to an 80th part of the whole, and which has been very inadequately canvassed.