may be considered as an elegant method of accounting for that order and proportion, which we every where see in the phænomena of nature. The determinate shapes, sizes, and mutual actions of the constituent particles of matter, fix the ratios between the causes for the happenings, and the failures; and therefore it is highly probable, and even necessary, as one may say, that the happenings and failures should perpetually recur in the same ratio to each other nearly, while the circumstances are the same. When the circumstances are altered, then new causes take place; and consequently there must be a new, but fixed ratio, between the happenings and the failures. Let the first circumstances be called A, the new ones B. If now the supposition be made so general, as equally to take in both A and B, the ratio of the happenings and failures will not be such as either A or B required. But still it will tend to a preciseness, just as they did, since the sum of the causes of the happenings must bear a fixed ratio to the sum of the causes of the failures.
An ingenious friend has communicated to me a solution of the inverse problem, in which he has shewn what the expectation is, when an event has happened p times, and failed q times, that the original ratio of the causes for the happening or failing of an event should deviate in any given degree from that of p to q. And it appears from this solution, that where the number of trials is very great, the deviation must be inconsiderable; which shews that we may hope to determine the proportions, and, by degrees, the whole nature, of unknown causes, by a sufficient observation of their effects.
The inferences here drawn from these two problems are evident to attentive persons, in a gross general way, from common methods of reasoning.
Let us, in the next place, consider the Newtonian differential method, and compare it with that of arguing from experiments and observations, by induction and analogy. This differential method teaches, having a certain number of the ordinates of any unknown curve given with the points of the absciss on which they stand, to find out such a general law for this curve, i.e. such an equation expressing the relation of an ordinate and absciss in all magnitudes of the absciss, as will suit the ordinates and points of the absciss given, in the unknown curve under consideration. Now here we may suppose the given ordinates standing upon given points to be analogous to effects, or the results of various experiments in given circumstances, the absciss analogous to all possible circumstances, and the equation afforded by the differential method to that law of action, which, being supposed to take place in the given circumstances, produces the given effects. And as the use of the differential method is to find the lengths of ordinates not given, standing upon points of the absciss that are given, by means of the equation, so the use of attempts to make general conclusions by induction and
