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argument in favour of causation * that all the planets move in the same direction. It would be no proof of asymmetry in a coin that it ever so often turned up in succession heads. Doubtless you never could prove by repeated throws the existence of such a peculiar kind of asymmetry, such a wabbling load, that it would be (for each throw) “as likely to bring the degree of success up to that point” which is observed, that is to give heads, as not to do so,” that is to give tails. Pure chance would always be as probable an hypothesis as that. In a word, Mr. Gurney's solution underrates the evidence in the case where the divergence from the most probable value is small or not known to be large, but is repeatedly in the same direction. In the general case where p is very small his solution does not differ substantially from
1 His 2p is as good as our - P, may be regarded as of the same order of magnitude.
It should be observed that this criticism relates to the second, not the first operation, as performed by Mr. Gurney. His scaffolding is more elaborate, if not more serviceable, than ours. But in the building he uses some materials which, though solid enough for ordinary purposes, yet will not bear certain strains. It is to be observed, also, that Mr. Gurney's “at least that degree of success” has here been interpreted as at least that degree of divergence from the most probable point in an assigned, say the plus, direction. If we interpret (violently) his q as probability of obtaining that degree of divergence in either direction, we shall be involved in still greater difficulties.
(3) As to our third problem, it has been already resolved into the other two.
III. We come now to the third, the calculative portion of our work. (1) As an example of the application of first principles without the intervention of approximative formulæ, let us take the experiment cited by Mr. Gurney at p. 251 of Part VII. of these Proceedings, where the name thought of” was DOREMOND, and the “letters
of a certain degree of success being attained may be put down as . The ground of my contention is that we are not entirely ignorant of the probability in question. For we have the datum that it is greater than the probability that chance alone would attain the certain degree of success. For it is absurd to suppose that chance + a favouring cause is less likely to obtain a certain degree of success than chance alone. Accordingly it might be legitimate to put q=p+'; or rather to regard yy as an independent variable in P, the expression for the à posteriori probability in favour of a cause, and to integrate P with regard to w between limits p and l; agreeably to the practice recommended by Donkin in his masterly discussion of a priori probabilities (Phil. Mag., 1851). It is clear that, when p is in the neighbourhood of 1, Mr. Gurney's assumption sacrifices much of the cumulative force which properly belongs to P.
* Cf. Laplace, Essai Philosophique.
produced were EPJYEIOD. Here, out of eight guesses, there are four successes ; if success consist in guessing either the very letter thought of, or either of its nearest alphabetical neighbours, in short any one of an assigned consecutive triplet. The probability that a letter taken at random should fall within any assigned triplet is g. Accordingly (on the supposition that chance is the only agency), the probabilities of obtaining in the course of eight trials no successes, one success, two successes, &c., are given by the first, second, third, &c., terms respectively of the binomial (&+$)%. The probability of obtaining at least four successes is equal to the sum of the fifth, and remaining terms; that is
70 (5) (3 )+56(3) (13)+28(3) () +8:(*) ()+(35;
or 011. The probability, then, in favour of an agency other than chance is about .99.
When larger numbers are involved, approximative formulæ become necessary. According to principles familiar to those who have studied the calculus of probabilities, the objective probability involved in either formula (a) or (b) is approximately*
The approximation requires that no should not exceed X, and that A should be large. This then, according to the reasoning employed in our second part, is the measure of the à posteriori probability in favour of chance alone.
For example, in the first instance given by Mr. Gurney, at p. 241 of the December number of this Journal, N is 2927, u is \, and v is
As the condition required for the validity of the approximation is just or very nearly fulfilled, the answer is, if I do the sum correctly, about .93 as the probability of an agency other than chance; no very crushing probability, as statistical evidence goes. In Mr. Gurney's next instance, N is 1833, u still \, v is 1533. Whence in favour of additional agency a very respectable probability, :997.
(2) and (3). As an illustration of the second problem (including the
Omitting a certain term outside the sign of integration (see Todhunter, sec. 997) as here practically, if not in general theoretically, neglectible. It will be observed that in halving the quantity within the brackets we assume that an excess greater than n is equally probable as a defect greater than the same quantity. This is exactly true only when u= In our case the factor is too large. The argument becomes a fortiori,
third), let us suppose that the series just instanced breaks up into four series, each presenting an excess of successes, with about the same van arrangement to which the experiments of M. Richet (described at p. 622-628 of the December number of Revue Philosophique) seem to lend themselves without violence. Then for one of the fractional series we have N=1933=458, u and v
Whence p is found about .08. Whence pt about .00004. And 1-p4, the measure of the sought probability, =:99996, which may fairly be regarded as physical certainty. It should be observed that if, as would usually happen, the v for all the partial series should not be the same, then ceteris paribus the above estimate would be below the mark. On the other hand, if the partial N's were unequal, ceteris paribus our estimate would be above the mark. As both inequalities, but especially the former, are likely to make themselves felt, the conclusion may be regarded
Such is the evidence which the calculus of probabilities affords as to the existence of an agency other than mere chance. The calculus is silent as to the nature of that agency-whether it is more likely to be vulgar illusion or extraordinary law. That is a question to be decided, not by formulæ and figures, but by general philosophy and common
ANNUAL BUSINESS MEETING.
The Third Annual Business Meeting of the Members of the Society was held at 14, Dean's Yard, London, S.W., on the 30th of January.
The President, Professor H. Sidgwick, briefly referred to the growth of the Society during the previous year. He remarked that during 1884 the Society nearly doubled the number of its Members and Associates. At the close of the year the Society consisted of :Members
258 Honorary Members
6 Corresponding Members
9 Honorary Associates
21 Vice-Presidents who are not Members or Associates
520 The President also referred to the Library as numbering nearly 800 volumes. Of these works, 520 are in English, 135 in German, 110 in French, and 15 in other languages.
An audited balance sheet of the receipts and expenditure of the Society during the year 1884 was placed before the Meeting. In commenting on it the President said it appeared that, after taking account of moneys due and owing at the end of the year, there was still a balance on the right side ; in addition to which there was the Library, the Stock of Proceedings, and the furniture and fittings belonging to the Society. It was agreed that a valuation of these should be made during the current year, so that at its close the Society might know its exact position both as to capital and as to receipts and expenditure.
The six vacancies on the Council, caused by the retirement in rotation of five Members, and by the death of Mr. Walter H. Browne, were filled by the election of the following gentlemen :G. P. Bidder, Q.C.
Rev. W. Stainton Moses.
C. Lockhart Robertson, M.D.
J. Herbert Stack. The approval of the Society was obtained to a change in the relations between the Council and the investigating Committees. future, the responsibility for both the facts and the reasonings in papers published in the Proceedings will rest entirely with their authors; and the Council, as a body, will refrain from expressing or implying any opinion on the subjects thus brought forward. The papers will, however, be submitted to a Committee of Reference before publication.
PROCEEDINGS OF THE GENERAL MEETINGS IN
May and June, 1885.
The fourteenth and fifteenth General Meetings of the Society were held at the Rooms of the Society of British Artists, Suffolk-street, Pall Mall, on Friday, May 29th, and Friday, June 24th.
Mr. F. W. H. MYERS IN THE CHAIR. The programme on both occasions included parts of Mr. Hodgson's account of his investigations in India, and of the paper on “Some Higher Aspects of Mesmerism,” which appear below. At the June
reeting Professor Sidgwick read the conclusions expressed by the Com mittee in the following Report.
REPORT OF THE COMMITTEE
INVESTIGATE PHENOMENA CONNECTED WITH THE
THEOSOPHICAL SOCIETY *
1. STATEMENT AND CONCLUSIONS OF THE COMMITTEE.
In May, 1884, the Council of the Society for Psychical Research appointed a Committee for the purpose of taking such evidence as to the alleged phenomena connected with the Theosophical Society as might be offered by members of that body at the time in England, or as could be collected elsewhere.
The Committee consisted of the following members, with power to add to their number :-Messrs. E. Gurney, F. W. H. Myers, F. Podmore, H. Sidgwick, and J. H. Stack. They have since added Mr. R. Hodgson and Mrs. H. Sidgwick to their number.
For the convenience of Members who may not have followed the progress of the Theosophical Society, a few words of preliminary explanation may be added here.
The Theosophical Society was founded in New York, in 1875, by Colonel Olcott and Madame Blavatsky, ostensibly for certain philanthropic and literary purposes. Its headquarters were removed to India in 1878, and it made considerable progress among the Hindus and other
As this Committee had carried out a large portion of its work before the appointment of the Committee of Reference, its Report has, by exception, not been submitted