the separate probability of each evidence is greater or less. Thus the principal facts of ancient history are not less probable practically now, than ten or fifteen centuries ago, nor less so then, than in the times immediately succeeding; because the diminution of evidence in each century is imperceptible. For, if be equal to 1, will be equal to 1 also; and if the deficiency of from 1 be extremely small, that of will be extremely small also, unless n be extremely great. And for the same reason a large number of weak arguments proves little; for the deficiency of each argument, being extremely great, , the resulting deficiency of independent evidences, will be extremely great also.
It appears likewise, that the inequality of the separate evidences does not much affect this reasoning. In like manner, if the number of evidences, dependent or independent, be great, we may make great concessions as to the separate values of each. Again, a strong evidence in dependent ones can add nothing, but must weaken a little; and, after a point is well settled by a number of independent ones, all that come afterwards are useless, because they can do no more than remove the imperceptible remaining deficiency, &c. And it will be of great use to pursue these and such like deductions, both mathematically, and by applying them to proper instances selected from the sciences, and from common life, in order to remove certain prejudices, which the use of general terms, and ways of speaking, with the various associations adhering to them, is apt to introduce and fix upon the mind. It cannot but assist us in the art of reasoning, thus to take to pieces, recompose, and ascertain our evidences.
If it be asked, upon what authority absolute certainty is represented by unity, and the several degrees of probability by fractions less than unity, in the doctrine of chances? also, upon what authority the reasoning used in that doctrine is transferred to other subjects, and made general, as here proposed? I answer, that no person who weighs these matters carefully, can avoid giving his assent; and that this precludes all objections. No sceptic would, in fact, be so absurd as to lay two to one, where the doctrine of chances determines the probability to be equal on each side; and therefore we may be sure, that he gives a practical assent at least to the doctrine of chances.
M. De Moivre has shewn, that where the causes of the happening of an event bear a fixed ratio to those of its failure, the happenings must bear nearly the same ratio to the failures, if the number of trials be sufficient; and that the last ratio approaches to the first indefinitely, as the number of trials increases. This
