This Method, in more extensive and complex cases, is a great aid to the calculator in his inferences from facts, and removes much that is arbitrary in the Method of Means.
Sect. IV.—The Method of Residues.
15. By either of the preceding Methods we obtain, from observed facts, such Laws as readily offer themselves; and by the Laws thus discovered, the most prominent changes of the observed quantities are accounted for. But in many cases we have, as we have noticed already, several Laws of nature operating at the same time, and combining their influences to modify those quantities which are the subjects of observation. In these cases we may, by successive applications of the Methods already pointed out, detect such Laws one after another: but this successive process, though only a repetition of what we have already described, offers some peculiar features which make it convenient to consider it in a separate Section, as the Method of Residues.
16. When we have, in a series of changes of a variable quantity, discovered one Law which the changes follow, detected its Argument, and determined its Magnitude, so as to explain most clearly the course of observed facts, we may still find that the observed changes are not fully accounted for. When we compare the results of our Law with the observations, there may be a difference, or as we may term it, a Residue, still unexplained. But this Residue being thus detached from the rest, may be examined and scrutinized in the same manner as the whole observed quantity was treated at first: and we may in this way detect in it also a Law of change. If we can do this, we must accommodate this new found Law as nearly as possible to the Residue to which it belongs; and 217 this being done, the difference of our Rule and of the Residue itself, forms a Second Residue. This Second Residue we may again bring under our consideration; and may perhaps in it also discover some Law of change by which its alterations may be in some measure accounted for. If this can be done, so as to account for a large portion of this Residue, the remaining unexplained part forms a Third Residue; and so on.
17. This course has really been followed in various inquiries, especially in those of Astronomy and Tidology. The Equation of the Center, for the Moon, was obtained out of the Residue of the Longitude, which remained when the Mean Anomaly was taken away. This Equation being applied and disposed of, the Second Residue thus obtained, gave to Ptolemy the Evection. The Third Residue, left by the Equation of the Center and the Evection, supplied to Tycho the Variation and the Annual Equation. And the Residue, remaining from these, has been exhausted by other Equations, of various arguments, suggested by theory or by observation. In this case, the successive generations of astronomers have gone on, each in its turn executing some step in this Method of Residues. In the examination of the Tides, on the other hand, this method has been applied systematically and at once. The observations readily gave the Semimensual Inequality; the Residue of this supplied the corrections due to the Moon’s Parallax and Declination; and when these were determined, the remaining Residue was explored for the law of the Solar Correction.
18. In a certain degree, the Method of Residues and the Method of Means are opposite to each other. For the Method of Residues extricates Laws from their combination, bringing them into view in succession; while the Method of Means discovers each Law, not by bringing the others into view, but by destroying their effect through an accumulation of observations. By the Method of Residues we should first extract the Law of the Parallax Correction of the Tides, and then, from the Residue left by this, obtain the Declination Correction. But we might at once employ the Method 218 of Means, and put together all the cases in which the Declination was the same; not allowing for the Parallax in each case, but taking for granted that the Parallaxes belonging to the same Declination would neutralize each other; as many falling above as below the mean Parallax. In cases like this, where the Method of Means is not impeded by a partial coincidence of the Arguments of different unknown Inequalities, it may be employed with almost as much success as the Method of Residues. But still, when the Arguments of the Laws are clearly known, as in this instance, the Method of Residues is more clear and direct, and is the rather to be recommended.
19. If for example, we wish to learn whether the Height of the Barometer exerts any sensible influence on the Height of the Sea’s Surface, it would appear that the most satisfactory mode of proceeding, must be to subtract, in the first place, what we know to be the effects of the Moon’s Age, Parallax and Declination, and other ascertained causes of change; and to search in the unexplained Residue for the effects of barometrical pressure. The contrary course has, however, been adopted, and the effect of the Barometer on the ocean has been investigated by the direct application of the Method of Means, classing the observed heights of the water according to the corresponding heights of the Barometer without any previous reduction. In this manner, the suspicion that the tide of the sea is affected by the pressure of the atmosphere, has been confirmed. This investigation must be looked upon as a remarkable instance of the efficacy of the Method of Means, since the amount of the barometrical effect is much smaller than the other changes from among which it was by this process extricated. But an application of the Method of Residues would still be desirable on a subject of such extent and difficulty.
20. Sir John Herschel, in his Discourse on the Study of Natural Philosophy (Articles 158–161), has pointed out the mode of making discoveries by studying Residual Phenomena; and has given several illustrations of the process. In some of these, he has also 219 considered this method in a wider sense than we have done; treating it as not applicable to quantity only, but to properties and relations of different kinds.
We likewise shall proceed to offer a few remarks on Methods of Induction applicable to other relations than those of quantity.