## Rmg

Fig. 5.6. Copernicus' theory for the superior planets from a geostationary point of view (solid lines) superimposed on elements of his heliocentric scheme (dashed lines).

Earth as shown in Figure 5.6, the planet is at EO' + O'C' + C'C" + C"P, the two being identical, since EO' = SO, O'C' = OC, C'C" = CP and C"P = ESS. So Copernicus' geometrical scheme is equivalent entirely to a geocentric model in which the planet moves on a double epicycle around an eccentric deferent or, since motion around an eccentric deferent can always be replaced by an epicycle, to a triple epicycle model like those of Ibn al-Shatir.

It is illuminating to compare Copernicus' theory with that of Ptolemy. The latter's success was based on his introduction of the equant and Copernicus wanted to achieve the effect of the equant without actually having one. If we construct the line C" Q, parallel to C O', intersecting the apsidal line at Q, elementary geometry implies that O'Q is constant with |O'Q| = |CC"\ = |CP|, and Z C" QA' = ¿COA and, hence, increases uniformly with time. Thus the point Q plays the role of the equant in Copernicus' theory and if C", which is the centre of the planet's epicycle, moved around O' in a circle, Copernicus' model would be identical geometrically to that of Ptolemy. It does not (though its path is almost circular given that |C'C"| is small relative to |OC'|) but if we were to try to match up with Ptolemy's model on the apsidal line, say, we would want to choose (remembering that Ptolemy has the centre of his deferent

Copernicus' planetary theory Table 5.1. The parameters in the models for the superior planets.

Ptolemy Copernicus

Copernicus' planetary theory Table 5.1. The parameters in the models for the superior planets.

Ptolemy Copernicus

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