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the proportion of the number of times it will happen, to the number of times it will fail in thoſe trials, ſhould differ leſs than by ſmall aligned limits from the proportion of the probability of its happening to the probability of its failing in one ſingle trial. But I know of no perſon who has ſhewn how to deduce the ſolution of the converſe problem to this; namely, “the number of times an unknown event has happened and failed being given, to find the chance that the probability of its happening ſhould lie ſomewhere between any two named degrees of probability.” What Mr. De Moivre has done therefore cannot be thought ſufficient to make the conſideration of this point unneceſſary: eſpecially, as the rules he has given are not pretended to be rigorouſly exact, except on ſuppofition that the number of trials made are infinite; from whence it is not obvious how large the number of trials muſt be in order to make them exact enough to be depended on in practice.

Mr. De Moivre calls the problem he has thus ſolved, the hardeſt that can be propoſed on the ſubjedt of chance. His ſolution he has applied to a very important purpoſe, and thereby ſhewn that thoſe miſtaken who have inſinuated that the Doctrine of Chances in mathematics is of trivial conſequence, and cannot have a place in any ſerious enquiry^{[1]}. The purpoſe I mean is, to ſhew what reaſon we have for believing that there are in the conſtitution of things fixt laws according to which events happen, and that, therefore, the frame of the world muſt be

- ↑ See his Doctrine of Chances, p. 252, &c.