There can be no doubt that the movements of the moon and of the planets would be, to a large extent, explained by such a system of circular orbits; but the spirit of astronomical enquiry is not satisfied with approximate results. Again and again the planets are observed, and again and again the observations are compared with the places which the planets would occupy if they moved in accordance with the system here indicated. The centres of the circles are moved hither and thither, their radii are adjusted with greater care; but it is all of no avail. The observations of the planets are minutely examined to see if they can be in error; but of errors there are none at all sufficient to account for the discrepancies. The conclusion is thus inevitable--astronomers are forced to abandon the circular motion, which was thought to possess such unrivalled symmetry and beauty, and are compelled to admit that the orbits of the planets are not circular. Then if these orbits be not circles, what are they? Such was the great problem which Kepler proposed to solve, and which, to his immortal glory, he succeeded in solving and in proving to demonstration. The great discovery of the true shape of the planetary orbits stands out as one of the most conspicuous events in the history of astronomy. It may, in fact, be doubted whether any other discovery in the whole range of science has led to results of such far-reaching interest. We must here adventure for a while into the field of science known as geometry, and study therein the nature of that curve which the discovery of Kepler has raised to such unparalleled importance. The subject, no doubt, is a difficult one, and to pursue it with any detail would involve us in many abstruse calculations which would be out of place in this volume; but a general sketch of the subject is indispensable, and we must attempt to render it such justice as may be compatible with our limits. The curve which represents with perfect fidelity the movements of a planet in its revolution around the sun belongs to that well-known group of curves which mathematicians describe as the conic sections. The particular form of conic section which denotes the orbit of a planet is known by the name of the _ellipse_: it is spoken of somewhat less accurately as an oval. The ellipse is a curve which can be readily constructed. There is no simpler method of doing so than that which is familiar to draughtsmen, and which we shall here briefly describe. We represent on the next page (Fig. 37) two pins passing through a sheet of paper. A loop of twine passes over the two pins in the manner here indicated, and is stretched by the point of a pencil. With a little care the pencil can be guided so as to keep the string stretched, and its point will then describe a curve completely round the pins, returning to the point from which it started. We thus produce that celebrated geometrical figure which is called an ellipse.