<p style='text-align: justify;'>In his <i>Disputationes contra Cremonensia deliramenta</i>, the last book printed before his untimely death in 1476, Regiomontanus offered a critique of the <i>Theorica planetarum communis</i>, a thirteenth-century textbook attributed to Gerard of Cremona, in comparison to the relative advantages offered by Georg Peuerbach's <i>Theoricae novae planetarum</i>. Adopting the form of a dialogue between 'Viennensis' (representing Regiomontanus) and 'Cracoviensis' (representing Martin Bylica of Ilkusch), the work utilises geometrical proofs, often supplemented by diagrams, to refute specific claims in the earlier <i>Theorica</i>. The image seen here was one of several diagrams used to criticise errors in the <i>Theorica planetarum communis</i> regarding the basic geometry of Ptolemy's Mercury model ' one of his most complex. Consistent with most of Ptolemy's planetary models, Mercury was said to revolve on the circumference of an epicycle, the centre of which was carried by an eccentric deferent. In order to fit the observational data, this deferent too had to be carried on a small eccentric deferent circle. Often referred to as a 'crank mechanism', this small circle carried on its circumference the centre of the large deferent, thereby cyclically moving it closer to and farther away from the Earth (represented at n). This image represents the motion of Mercury according to the counter-clockwise movement of its eccentric deferent and the clockwise motion of the small inner circle. It essentially offers two snapshots of the model, superimposed to show how the relative positions of the circles have changed as they rotate through ninety degrees. The centre of the epicycle of Mercury is represented in its original position on the eccentric deferent at a; as the deferent carries the epicycle uniformly and counter-clockwise about the equant point g, the small circle moves the centre of the deferent in a clockwise direction at the same rate (from e to m, about the centre f). Thus, the angles of the inner circle and the deferent change uniformly, but in opposite directions. As a result of this dual motion, the centre of deferent moves from e to m, while the centre of the epicycle has moved from a to k. Using this diagram, Regiomontanus refutes an excessive generalisation about uniform motion in the <i>Theorica planetarum communis</i>, in which it is stated that as the centre of Mercury's deferent is carried clockwise from e to m about f, so <i>any point</i> on the circumference of the deferent moves uniformly at the same rate counter-clockwise about the equant. By referring to c, the point opposite the epicycle centre a on diameter ac, Regiomontanus shows this to be false. As the circles rotate through ninety degrees, diameter ac now lies at kl; point c has demonstrably swept out more than ninety degrees about equant g. Thus Regiomontanus proves that the behaviour of the epicycle centre about the equant, as it moves from a to k, cannot be generalised to other points of the eccentric circle, not even to its corresponding opposite point on the diameter.</p>
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