Ellipse

“The planetary orbits are not exact circles, but are more or less elliptical, the Sun being situated not at the centre, but at one of the foci of the ellipse. The distance of either focus from the centre indicates the eccentricity of the ellipse. On account of this eccentricity the distance of a planet from the Sun, and its velocity of revolution, vary in different parts of its orbit.”

“Kepler's observation of the elliptical rotation of the planets was the first of three laws, quantitatively expressed, which paved the way for Newton's law. Why did the planets move in just this way? Kepler tried to answer this also, but failed. It remained for Newton to supply the answer to this question.”

“Since the ellipse is the most important of all conic sections in the study of the orbits of the planets, we discuss these curves in more detail, noting that the circle is a special case of an ellipse or, put differently, the ellipse is a symmetrical distortion of a circle.”

“How can any prediction of the return of a comet be made? and, supposing such a prediction to be possible, How can it thence be shown that this comet of 1835, is the same with those seen in 1682, in 1607, in 1531, and in 1456?Before the first of these questions is answered, the reader must be made acquainted with the nature and properties of a certain curve, called by geometers an ellipse.”

“If now the point H be brought nearer to S than it was before, the length of the string remaining the same, and the curve be then described as before, it will be found to have altered its form, so as more nearly to approach that of a circle, as shown in the... figure, and H may be brought so near to S that its form shall scarcely be distinguishable from that of a circle, which figure it would indeed manifestly become, if the point H were made accurately to coincide with S.”