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 figure lasted without the smallest alteration until it began to return towards the sun where it leaves the tangent drawn to its epicycle. In the Ptolemaic system the earth's centre was regarded as the centre of the universe, and the movements of the heavenly bodies were explained by eccentrics and epicycles. The sun was conceived to describe a circle about a point not exactly coinciding with the centre of the earth, called the sun's eccentric. The planets described epicycles (circles) whose centres described eccentrics (circles), and the centres of these eccentrics coincided with the centre of the sun's eccentric. In the case of Mercury and Venus the centre of the epicycle was always on the line drawn from the centre of the eccentric to the sun's centre. In the case of the other planets the construction was more complicated. The stationary points were determined by drawing tangents from the earth's centre (or the observer) to the epicycle, as in the figure (1).—(, Institutio Astronomica, 1647.) This will explain Kepler's description of the stationary points as the points where the planet leaves the tangent to its epicycle, supposing that he uses the terms of the current (i.e. Ptolemaic) astronomy. Copernicus placed the sun instead of the earth at the centre of the universe, but to determine the positions of the planets at any given time with as much accuracy as was attainable with the Ptolemaic system, he was obliged to use a similar method of eccentrics and epicycles, so that Kepler's expression may be understood to describe the stationary points according to the Copernican theory, though it is still strange that he should not recognise the elliptical form of the planetary orbits, which he had lately demonstrated after most laborious reasoning in his Commentaries on the Motion of the Planet Mars, 1609. Galileo's own expression seems to describe the stationary points according to the Copernican system, as would be expected, as the points where the planet leaves the tangent drawn to its orbit from the earth (Fig. 2). At this time it loses the semicircular form more and more, and keeps on diminishing that figure until its conjunction, when it will wane to a very thin crescent. After completing its passage past the sun, it will appear to us, at its