The facts, that is what was done and written, are of course correct; but the conclusions drawn from them are highly controversial to the present day.
The calculations for finding an unknown planet by the perturbations it causes in the orbit of another are extremely difficult, the more so when the data are small and uncertain. For Percival they were very small because Neptune,—nearest to the unknown body,—had been discovered so short a time that its true orbit, apart from the disturbances therein caused by other planets, was by no means certain. In fact Percival tried to analyze its residuals, but they yielded no rational result. This left only what could be gleaned from Uranus after deducting the perturbations caused by Neptune, and that was small indeed. In 1845, when the calculations were made which revealed that planet, “the outstanding irregularities of Uranus had reached the relatively huge sum of 133″. To-day its residuals do not exceed 4.5″ at any point of its path.”
Then there are uncertainties depending on errors of observation, which may be estimated by the method of least squares of the differences between contemporary observations. Moreover there is the uncertainty that comes from not knowing how much of the observed motion is to be attributed to a normal orbit regulated by the Sun, and how much to the other planets, including the unknown. Its true motion under these influences can be ascertained only by observing it for a long time, and by taking periods sufficiently far apart to distinguish the continuing effects of the known bodies from those that flow from an unknown source. This was the ingenious method devised by Leverrier as a basis for his calculations, and he thereby got his residuals caused by the unknown planet in a form that could be handled.
Finally there was the uncertainty whether the residual perturbations, however accurately determined, were caused by one or more outer bodies. Of this Percival was, of course, well aware, and in fact, in his study of the comets associated with Jupiter he had pointed out that there probably was a planet far beyond the one for which he was now in search. But, as no one has ever been able to devise a formula for the mutual attraction of three bodies, he could calculate only for a single body that would account as nearly as possible for the whole of the residuals.
Thus he knew that his work was an approximation; near enough, he hoped, to lead to the discovery of the unknown.
The various elements in the longitude of a planet’s orbit, that is in the plane of the ecliptic, that are affected by and affect another, are:
a—The length of its major, or longest, axis.
n—Its mean motion, which depends on the distance from the Sun.
ε—The longitude at a given time, that is its place in its orbit.
e—The eccentricity of its orbit, that is how far it is from a circle.