of Mercury and Venus to be on a bar passing through the sun, and to be between the earth and the sun. The centres of the epicycles of Mars, Jupiter, and Saturn were supposed to be further away than the sun. Mercury and Venus were supposed to revolve in their epicycles in their own periodic times and in the deferent round the earth in a year. The major planets were supposed to revolve in the deferent round the earth in their own periodic times, and in their epicycles once in a year. It did not occur to Ptolemy to place the centres of the epicycles of Mercury and Venus at the sun, and to extend the same system to the major planets. Something of this sort had been proposed by the Egyptians (we are told by Cicero and others), and was accepted by Tycho Brahe; and was as true a representation of the relative motions in the solar system as when we suppose the sun to be fixed and the earth to revolve. The cumbrous system advocated by Ptolemy answered its purpose, enabling him to predict astronomical events approximately. He improved the lunar theory considerably, and discovered minor inequalities which could be allowed for by the addition of new epicycles. We may look upon these epicycles of Apollonius, and the excentric of Hipparchus, as the responses of these astronomers to the demand of Plato for uniform circular motions. Their use became more and more confirmed, until the seventeenth century, when the accurate observations of Tycho Brahe enabled Kepler to abolish these purely geometrical makeshifts, and to substitute a system in which the sun became physically its controller. FOOTNOTES: [1] _Phil. Mag_., vol. xxiv., pp. 481-4. [2] Plaeiadas t’ esoronte kai opse duonta bootaen ‘Arkton th’ aen kai amaxan epiklaesin kaleousin, ‘Ae t’ autou strephetai kai t’ Oriona dokeuei, Oin d’ammoros esti loetron Okeanoio. “The Pleiades and Boötes that setteth late, and the Bear, which they likewise call the Wain, which turneth ever in one place, and keepeth watch upon Orion, and alone hath no part in the baths of the ocean.” [3] See Pearson in the Camb. Phil. Soc. Proc., vol. iv., pt. ii., p. 93, on whose authority the above statements are made. [4] See p. 6 for definition. 4. THE REIGN OF EPICYCLES—FROM PTOLEMY TO COPERNICUS. After Ptolemy had published his book there seemed to be nothing more to do for the solar system except to go on observing and finding more and more accurate values for the constants involved--viz., the periods of revolution, the diameter of the deferent,[1] and its ratio to that of the epicycle,[2] the distance of the excentric[3] from the centre of the deferent, and the position of the line of apses,[4] besides the inclination and position of the plane of the planet’s orbit. The only object ever aimed at in those days was to prepare tables for predicting the places of the planets. It was not a mechanical problem; there was no notion of a governing law of forces.