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between the Sun and the Earth-Moon system over the course of 1 year.

However, this theory had the defect that the ratio of the greatest and least distances of the Moon did not correspond to the observed changes in apparent diameter. Ibn al-Shatir had managed to bring this ratio more into line with reality from that inherent in Ptolemy's theory, but it was still too large. The modification that was made to Tycho's model actually was performed by his long-serving assistant, Longomontanus, and consisted essentially of replacing the larger of Ibn al-Shatir's epicycles, of radius r\, by two smaller ones of radii 2ri /3 and r1 /3. The resulting set of epicycles had to be reshuffled so as to get the scheme to work, but the final theory resulted in very accurate displacements of about 40' 30'' in the position of the Moon at the octants, and the ratio of the greatest to least distances between the Earth and the Moon had been reduced from the Copernican value of 1. 31 to the much more accurate 1.16(1.141 being the modern value).32

Tycho also considered the problem of the Moon's latitude, and observed that the angle at which the Moon's orbit was inclined to the ecliptic was not the constant 5° that everyone assumed.33 Observations in 1587 had led him to believe that the inclination of the orbit of the Moon actually was 51 °, but in 1595 his observations indicated that the orbital plane actually varied between 5 and 5i ° during a synodic month. He postulated a small nutation, reminiscent of earlier theories of trepidation, with a period of half a synodic month; this is illustrated (greatly exaggerated) in Figure 6.4. At a conjunction or opposition, the pole of the orbit of the Moon is at P1 and the inclination of the orbit is 5°, whereas at quadrature this pole is at P3 and the inclination is 51 °. This latitude theory predicts an oscillation in the nodes of the lunar orbit: as the pole of the orbit of the Moon moves around the small circle P1, P2, P3, P4 the ascending node, which is B1 at syzygy, oscillates through the points B1, B2, B3, B4 along

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