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solving the triangle CEM.

The overall effect of Ptolemy's scheme was to leave the longitude at conjunc-tionand oppositionunchangedfrom that given by Hipparchus (since/ CEA = 0 in these cases) and the theory accounted well for the observed position of the Moon at the quadratures. However, Ptolemy found that there were still noticeable errors at the octants - the points midway between the quadratures and the syzygies - and decided that a further modification was necessary. He did this by making the Moon rotate around its epicycle at a non-uniform rate with respect to ECB, but uniform with respect to NCB', where the point N is such that E is the midpoint of ON. For this new scheme, it is the angle B'CM that is a given linear function of time, so in order to compute the prosthaphaeresis angle, Ptolemy first had to calculate /BCB' = LNCE. This he could do by solving the triangle NCE once the length | EC| had been determined. This final peculiarity in Ptolemy's lunar theory, which has no effect on the position of the Moon at the syzygies or at quadrature and which was another source of criticism by later astronomers, is known as prosneusis. In actual fact, the effect of prosneusis was to make things better some of the time but worse at others. Ptolemy did not test his final theory against new observations away from the

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