analogy, from particular effects or phænomena, in different given circumstances, by applying the general law conclusion to these circumstances.
This parallel is the more pertinent and instructive, inasmuch as the mathematical conclusion drawn by the differential method, though formed in a way that is strictly just, and so as to have the greatest possible probability in its favour, is, however, liable to the same uncertainties, both in kind and degree, as the general maxims of natural philosophy drawn from natural history, experiments, &c.
If many ordinates be given; if the distances of the points of the absciss, on which they stand, be equal and small; if the ordinate required lie amongst them, or near them; and if there be reason to think, that the curve itself is formed according to some simple, though unknown law; then may we conclude, that the new ordinate, determined by the equation, does not vary far from the truth. And if the resulting equation be simple, and always the same, from whatever given ordinates it be extracted, there is the greatest reason to think this to be the real original law or equation of the curve; and consequently that all its points and properties may be determined with perfect exactness by means of it: whereas, if the given ordinates be few, their distances great or unequal, the ordinate required considerably distant from many or most of them, the unknown curve be a line drawn at hazard, and the resulting equation different, where different ordinates are given, though their number be the same, there will be little probability of determining the new ordinate with exactness; however, still the differential method affords us the greatest probability which the data permit in such cases.
In like manner, if the experiments or observations be many, their circumstances nearly related to each other, and in a regular series, the circumstances of the effect to be investigated nearly related to them; also, if the real cause may be supposed to produce these effects, by the varieties of some simple law, the method of induction and analogy will carry great probability with it. And if the general conclusion or law be simple, and always the same, from whatever phænomena it be deduced, such as the three laws of nature, the doctrines of gravitation, and of the different refrangibility of light; or to go still higher, by taking a mathematical instance, the law for finding the coefficients of the integral powers of a binomial, deduced from mere trials in various powers; there can scarce remain any doubt, but that we are in possession of the true law inquired after, so as to be able to predict with certainty, in all cases where we are masters of the method of computation, or applying it; and have no reason to suspect, that other unknown laws interfere. But, if the given phænomena be few, their circumstances very different from each other, and from those of the effect to be predicted; if there be reason to suppose, that many causes concur in the producing
