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cauſe or action, one may make a judgment what is likely to be the conſequence of it another time, and that the larger number of experiments we have to ſupport a concluſion, ſo much the more reaſon we have to take it for granted. But it is certain that we cannot determine, at leaſt not to any 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 one who would give a clear account oſ the ſtrength of *analogical* or *inductve 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 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*^{[1]}, 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,

- ↑ See Mr. De Moivre’s
*Doctrine of Chances*, p. 243, &c. He has omitted the demonſtrations of his rules, but theſe have been ſince ſupplied by Mr. Simpſon at the concluſion of his treatiſe on*The Nature and Laws of Chance*.