=The Mathematical Astronomers.= After enthusiastically canvassing the traditional theories of antiquity, the Humanists had been unable to find one which would explain and organize the newly accumulated materials of their “new world.” But working in more or less narrow circles, natural science had already made a beginning in the midst of the Humanists. Beginning with Copernicus, an interest in physics and astronomy had been aroused, but in these early days it was more speculative than empirical. The speculations of the astronomers had but little influence upon their own time. However, when the ancient theories proved inadequate to explain the facts of the “new world,” and especially when the empirical researches of Galileo confirmed the speculations of his predecessors, the Renaissance turned away from antiquity to nature herself for an explanation. This was about the year 1600, the year of the beginning of the Natural Science period. The most prominent of these astronomers were―― Copernicus, 1473–1543, a Pole. Bruno, 1548–1600, an Italian. Tycho Brahe, 1546–1601, a Dane. Kepler, 1571–1630, a German. Galileo, 1564–1641, an Italian. Huyghens, 1629–1695, a Hollander. Newton, 1642–1722, an Englishman. While the greatest of these scientists is Newton, who belongs to the next period, the most influential is Galileo. Modern _methods in science_ began with Galileo. Of the four predecessors of Galileo three――Copernicus, Tycho Brahe, and Bruno――are in spirit Humanists; for their final explanation of nature is the world of spirits. Kepler belongs to both the Humanistic and Natural Science periods; for at first he constructed his natural science by an amalgamation of the doctrine of spirits and the Copernican theory; but in the latter part of his life he adopted completely the mechanical view of nature. The above scientists may be divided for convenience into two groups: (1) the speculative scientists before Galileo; (2) Galileo and the following empirical investigators. For fourteen centuries the ancient Ptolemaic astronomy had been regarded by the learned as beyond question. Although complex and unwieldy, it explained all phenomena satisfactorily enough as they appeared to the senses; and it brought phenomena into a system. (The Ptolemaic system has been fully described in vol. i, pp. 322 ff.) To recapitulate it: the world-all was conceived as a hollow sphere with the earth as the centre and the fixed stars in the periphery, while the planets were supposed to move in epicycles. The universe was divided into the heavenly and terrestrial realms, which were occupied by various spirits. God resided outside this hollow sphere and held it, as it were, in his lap.