The primary purpose of describing planetary orbits in terms of epicycles and deferents was to account for the

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The question asks about why ancient models used epicycles and deferents to describe planetary orbits. 
Option 1: 'generally eastward motion of a planet compared to background stars while the whole sky appeared to move westward.' This describes the observable tendency of planets to drift east relative to stars, but it does not capture the key complication these models were designed to explain, namely the occasional backward motion. Therefore this option is incomplete as the primary purpose. 
Option 2: 'variation of brightness of a planet with time.' Brightness changes (like phases or brightness variations) are not what epicycles were meant to explain; the brightness is more related to intrinsic luminosity, distance, and reflectivity, not the geometric retrograde pattern, so this is not correct. 
Option 3: 'the differences in the speeds of the planets as they orbited the Earth.' While speed differences exist, epicycles and deferents were primarily introduced to account for retrograde motion, not just general speed differences, so this is mischaracterizing the motivation. 
Option 4: 'pattern of direct and retrograde motion of a planet as it moved slowly against the background of stars.' This precisely captures the central observational puzzle that epicycles and deferents were created to model: the apparent direct (prograde) motion and the periodic retrograde motion relative to the stellar backdrop. This option aligns with the historical purpose of the geocentric epicycle model. 
In summary, by process of elimination, the first three options fail to address the core observational feature that motivated epicycles and deferents, while the fourth option directly names that pattern of direct and retrograde motion, which the model sought to explain.

2261 ASTRON 0088 SEC1070 Homework 5 - The Greeks

The primary purpose of describing planetary orbits in terms of epicycles and deferents was to account for the

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