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.