The mathematical account of the movements of the moon has its history. As we have seen, it was early realized that she revolved round and near the earth in a circular orbit. Soon it appeared that there were irregularities in this movement. The ‘First Anomaly’, a difference of speed observed at different parts of the orbit, was well understood by Hipparchus. It could be expressed, so as to ‘save the phenomena’, by either of two methods, both resting on the assumption that no curve except a circle was admissible, and both superseding the ingenious but cumbrous arrangement of ‘concentric Spheres’ known to Aristotle. One was that of ‘movable eccentrics’, where the orbit of the planet was round a point outside the earth, itself shifting. The other, which prevailed, and was finally adopted by Ptolemy, was that of epicycles, circles described round points in the primary orbit, by means of which the planet’s motion could be retarded or quickened at will, and its position modified. By this device, the visible _movement_ could be, and was, recorded with great accuracy, but sometimes at the expense of physical truth. Thus the epicyclic arrangement for the moon’s orbit involved, if closely looked into, the consequence that her distance from us at nearest must be half that at the farthest, and her angular diameter double! Kepler, after the work of a lifetime (1571-1630), discovered the cause of this ‘anomaly’ in the shape of the orbit, which is elliptical, not circular, and substituted ‘eccentricity’ for ‘anomaly’ as the key-word. Newton (1642-1727) proved that a body revolving round another _must_ move in an ellipse, with the larger body at one focus. Thus the wheel had come full circle, and physical and mathematical inquiry met after two thousand years of separation. The ‘Second Anomaly’ due to the action of the sun (the ‘Evection’) was indicated by Hipparchus, worked out as a phenomenon by Ptolemy, and its physical cause explained by Newton. The inclination of the moon’s path to the sun’s was known to Hipparchus as 5°, and the recession of her nodes was familiar to him. A third anomaly now known as ‘Variation’ is instructive because its discovery has been claimed for an Arabian astronomer of about A.D. 1000. After an exhaustive discussion during the last century (1836-71), it seems to be proved that the claim rested upon a mistake, and that the sole credit is due to Tycho Brahe (see Dreyer, p. 252). In fact, whatever in astronomy does not belong to modern science is Greek, after allowing for what the Greeks may have learnt in early ages from Chaldaeans or Egyptians. The Romans contributed nothing, the Indians learnt much from scientific men who accompanied Alexander, and used it skilfully, but did not advance it. And the modern makes a really continuous whole with the ancient Greeks, for it is not only astronomy which should be considered, but the essential preliminaries, such as the study of the Conic Sections, which, in its geometrical form, is purely