X. A new method of deducing a first approximation to the orbit of a comet from three geocentric observations

Comets are distinguished from the planets not only by the peculiarities that immediately strike the eye, but likewise by the circumstances attending their motion in the heavens. All the planets move round the sun in orbits nearly circular; they never deviate far from the ecliptic on either side; and they move in a manner not extremely irregular, and in one direction, according to the order of the signs in the zodiac. Comets, on the contrary, when they first come into view, assume gradually greater degrees of brightness, which they again lose by like gradations, and then disappear; thus seeming to visit the neighbourhood of the sun for a short time only, after which they retire into the immensity of space: they are seen in all quarters of the heavens: and their motion is exceedingly various and irregular; confined to no direction; sometimes greatly curved, and often hardly distinguishable from a rectilineal course. If, to phenomena so dissimilar, we add the prejudice which almost universally prevailed, that comets have only a temporary existence, and are produced by occasional causes, we shall not perhaps have much reason to be surprised that the true account of those bodies, which represents them as forming a part of the same system with the planets, eluded the sagacity of Kepler, to whom we are indebted for the first accurate knowledge of the laws of the planetary motions. This step in our knowledge of the universe was reserved for Sir Isaac Newton. The principal and leading discovery of that great philosopher consisted in generalizing the laws of Kepler; in proving that they are necessary consequences of a more general fact, namely, that all the planets are continually deflected from a rectilineal motion towards the sun in the inverse proportion of the squares of their distances from that body. He demonstrated that the motions of such a system of bodies must be performed in the conic sections, having the sun in the focus, the species of the curve depending upon the proportion of the rectilineal velocity to the quantity of the deflection towards the common centre. This theory comprehends an infinite variety of motions, all flowing from one common principle; and the ellipse alone, by the changes of form which it undergoes according to the degrees of its eccentricity, seems, at one extreme, when it is greatly elongated, as well adapted to account for the phenomena of the comets, as it is, at the other extreme, when it differs little from a circle, to represent the motions of the planets.