any nicety, in what degree repeated experiments confirm a conclusion, without the particular discussion of the beforementioned problem; which, therefore, is ecessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning; concerning, which at present, we seem to know little more than that it does sometimes in fact convince us, and at other times not; and that, as it is the means of cquainting us with many truths, of which otherwise we must have been ignorant; so it is, in all probability, the ource of many errors, which perhaps might in some measure be avoided, if the force that this sort of reasoning ought to have with us were more distinctly and clearly understood These observations prove that the problem enquired after in this essay is no less important than it is curious. It may be safely added, I fancy, that it is also a problem that has never before been solved. Mr. De moivre, indeed, the great improver of this part of mathematics, has in his Laws of chance*, after Bernoulli, and to a greater degree of exactness, given rules to find the probability there is, that if a very great number of trials be made concerning any event the proportion of the number of times it will happen, to the number of time it will fail in those trials, should differ less than by small assigned limits from the proportion of its failing in one single trial. But I know of no person who has shown 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 rigorously exact, except on supposition that the number of trials are made 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. Ir. De Moivre calls the problem he has thus solved, the hardest that can be proposed on the subject of chance. His solution he has applied to a very important purpose, and thereby shewn that those a remuch mistaken who have insinuated that the Doctrine of Chances in mathematics is of trivial consequence and cannot have a place in any serious enquiry. 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 things happen, and that, therefore, the frame of the world must be the effect of the wisdom and power of an intelligent cause and thus to confirm the argument taken from final causes for the existence of he Deity. It will be easy to see that the converse problem solved in this essay is more directly applicable to this purpose; for it shews us, with distinctness and ery case of reason there is to think that such recurrency or order is derived from stable causes or regulations innature, and not from any irregularities of chance. c. He has omitted the demonstration of his rules, but these have been supplied by Mr. Simpson at the conclusion of his treatise on The Nature and Laws of Chance. tSee his Doctrine of Chances, p. 252, &eany nicety, in what degree repeated experiments confirm a conclusion, without the particular discussion of the beforementioned problem; which, therefore, is necessary to be considered by any that would give a clear account of the strength of analogical or inductive reasoning; concerning, which at present, we seem to know little more than that it does sometimes in fact convince us, and at other times not; and that, as it is the means of cquainting us with many truths, of which otherwise we must have been ignourant; so it is, in all probability, the source of many errors, which perhaps might in some measure be avoided, if the force that this sort of reasoning ought to have with us were more distinctly and clearly understood. These observations prove that the problem enquired after in this essay is no less important than it is curious. It may be safely added, I fancy, that it is also a problem that has never before been solved. Mr. De Moivre, indeed, the great improver of this part of mathematics, has in his Laws of chance∗, after Bernoulli, and to a greater degree of exactness, given rules to find the probability there is, that if a very great number of trials be made concerning any event, 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 assigned limits from the proportion of its failing in one single trial. But I know of no person who has shown 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 supposition that the number of trials are made 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 subject of chance. His solution he has applied to a very important purpose, and thereby shewn that those a remuch mistaken who have insinuated that the Doctrine of Chances in mathematics is of trivial consequence, and cannot have a place in any serious enquiry†. 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 things happen, and that, therefore, the frame of the world must be the effect of the wisdom and power of an intelligent cause; and thus to confirm the argument taken from final causes for the existence of the Deity. It will be easy to see that the converse problem solved in this essay is more directly applicable to this purpose; for it shews us, with distinctness and precision, in every case of any particular order or recurrency of events, what reason there is to think that such recurrency or order is derived from stable causes or regulations innature, and not from any irregularities of chance. ∗See Mr. De Moivre’s Doctrine of Chances, p. 243, &c. He has omitted the demonstration of his rules, but these have been supplied by Mr. Simpson at the conclusion of his treatise on The Nature and Laws of Chance. †See his Doctrine of Chances, p. 252, &c. 2