we have the means of observing are small in comparison with the total quantities, there is much danger lest we should mistake the numerical law, and be led to miscalculate the variations which would take place beyond the limits; a miscalculation which would vitiate any conclusion respecting the dependence of the effect upon the cause, that could be founded on those variations. Examples are not wanting of such mistakes. “The formulæ,” says Sir John Herschel,1 “which have been empirically deduced for the elasticity of steam (till very recently), and those for the resistance of fluids, and other similar subjects,” when relied on beyond the limits of the observations from which they were deduced, “have almost invariably failed to support the theoretical structures which have been erected on them.”
In this uncertainty, the conclusion we may draw from the concomitant variations of a and A, to the existence of an invariable and exclusive connection between them, or to the permanency of the same numerical relation between their variations when the quantities are much greater or smaller than those which we have had the means of observing, can not be considered to rest on a complete induction. All that in such a case can be regarded as proved on the subject of causation is, that there is some connection between the two phenomena; that A, or something which can influence A, must be one of the causes which collectively determine a. We may, however, feel assured that the relation which we have observed to exist between the variations of A and a, will hold true in all cases which fall between the same extreme limits; that is, wherever the utmost increase or diminution in which the result has been found by observation to coincide with the law, is not exceeded.
The four methods which it has now been attempted to describe, are the only possible modes of experimental inquiry—of direct induction a posteriori, as distinguished from deduction: at least, I know not, nor am able to imagine any others. And even of these, the Method of Residues, as we have seen, is not independent of deduction; though, as it also requires specific experience, it may, without impropriety, be included among methods of direct observation and experiment.