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338 at the syzygies and quadratures, but followed her throughout her monthly course year after year, determining her position both on and off the meridian, and not forgetting to observe her at apogee, or, as he called it, "in maxima remotione utriusque epicycli." He thus succeeded in detecting the third inequality in the motion in longitude, the variation, which reaches its maximum of 39′.5 (Tycho found 40′.5) in the octants, when the difference of longitude of sun and moon is 45°, 135°, &c. But apart from this, he could not be satisfied with the way in which Ptolemy had represented the motion in longitude (by a deferent and one epicycle, the centre of the former moving in a circle round the earth in a retrograde direction), because it represented the apparent diameters of the moon very badly. In fact, the moon ought, according to the theory of Ptolemy, to appear nearly twice as great at perigee as at apogee. This had not escaped Copernicus, who avoided it by making the deferent concentric with the earth, and adding a second epicycle with a motion twice as rapid as the first one. Tycho chose another way of representing the motion in longitude. The deferent (radius = 1) according to him had its centre in a circle with radius 0.02174, in the circumference of which the earth was placed, so that the centre of the deferent was in the earth in the syzygies, and farthest from it at the quadratures. There were two epicycles with radii 0.058 and 0.029, the period in the former being the anomalistic month, and the moon moving in the latter twice as rapidly and in the opposite direction, in such a manner that at apogee the moon was 0.029 outside the deferent, at perigee 0.058 + 0.029 = 0.087 inside it. The effect of the two epicycles gave the maximum of the first inequality 4° 59′ 30″, while the circle through the earth gave