Cambridge Philosophical Society. 151 



instance only of the predicate ; and shows that the removal of this 

 usual restriction entirely removes all his objections to Sir William 

 Hamilton's form of his own system. 



Section VI. On the application of the theory of probabilities to some 

 questions of evidence. — This inquiry was suggested by the apparent 

 (but only apparent) error of the logicians, who seem to lean towards 

 the maxim that, when the subject and predicate are unknown, the 

 universal and particular propositions ' Every X is Y,' ' Some Xs are 

 not Ys,' are a priori of equal probability. The difficulty is one which 

 occurs in the following case : — If a good witness, drawing a card 

 from a pack, were to announce the seven of spades, his credit would 

 not be lowered, though he would have asserted an event against which 

 it was 51 to 1 « priori. A common person gives the true answer, 

 * Why not the seven of spades as well as any other ? ' Many readers 

 of works on probability would be inclined to say ' That is not the 

 question ; why the seven of spades rather than some one or another 

 of the fifty-one others ? ' The retort is fallacious : it rubs out the 

 distinctive marks from the other fifty-one cards, and writes on each 

 of them ' not the seven of spades ' as its only exponent. Laplace 

 has chosen two problems, in one of which the distinctive marks 

 exist, and not in the other ; and, neglecting the consideration of the 

 first one, has founded his remarks upon the deterioration of evidence 

 by the assertion of an improbable event, entirely upon the second. 

 The object of this section is, by a closer examination of the mathe- 

 matical problem of evidence, to ascertain the accordance or non- 

 accordance of the results of usual data with usual notions. The 

 result of the examination is, that common notions, as in other cases, 

 are found closely accordant with theory. For instance, if there be 

 n possible things which can happen, so that the mean probability of 



an event is — , a witness of whom we know no particular bias towards 

 n 



one mode of error rather than another, asserting an event of which 



the a priori probability is a, has his previous credit raised, unaltered, 



or lowered, according as a— — is positive, nothing, or negative. So 



that though the a, priori probabilities were distributed among a mil- 

 lion of possible and distinguishable cases, yet a witness asserting one 

 of them against which it is only 999,999 to 1, would have as good 

 a right to be believed as though there had been but two equally pro- 

 bable cases, of which he had asserted one. 



March 11. — Curvature of Imperfectly Elastic Beams. By Ho- 

 mersham Cox, B.A. Jesus College. 



The equation to the curve of an elastic deflected beam is usually 

 deduced from the assumption, — 1, that the longitudinal compression 

 or extension of an elastic filament is proportional to the compressing 

 or extending force ; 2, that for equal extension and compression the 

 compressing and extending forces are equal to each other. 



These hypotheses are not quite correct in practice. All substances 

 appear to be subject to a defect of elasticity, i. e. their elastic forces 

 of restitution increase in a somewhat less degree than in proportion 

 to the extension or compression. If the forces be taken as functions 



