Three observations are necessary for the calculation of a parabolic orbit – Cometary ephemerides; what is meant by an ephemeris; control afforded by the ulterior observations –Elements of an elliptic orbit – Can the apparition or return of a comet be predicted?– State of the question – Refutation by Arago of a current prejudice

Three observations of a comet–that is to say, three different positions (in right ascension and declination) of the nucleus of a comet, or, in a word, three points of its trajectory or apparent orbit sufficiently distant from each other–are required, as we have said, for the calculation of the parabolic elements of the true orbit.

In the last century this determination was not only a long and laborious operation, but involved much tentative and uncertain work. Before engaging in the difficult calculation of the elements of an orbit, astronomers made trial graphically and even mechanically of different parabolas, and only began the calculation after satisfying themselves that one of these curves nearly represented the positions furnished by observation. Great improvements were introduced into these methods during the last century by Lalande, Laplace, and Gauss. But the calculation of a cometary orbit is always a sufficiently complex operation, even if it be simply parabolic, and it still takes a skilful computer accustomed to this kind of work, several hours to find approximate values of the different elements. This is not the place for us, of course, to attempt an explanation of the work itself.