Ill
29| for Saturn ; while in each case the epicycle was describedby the planet in one year, and in such a manner that the radiusfrom its centre to the planet, viz., pm, or pj, or ps, would pointdirectly towards the Earth , whenever the planet was seen in a,direction exactly opposite to that of the Sun (or, as it is termed,in Opposition), and directl yfrom the Earth , whenever the planetwas seen in the same direction as the Sun (or, as it is termed,in Conjunction).
It also followed, as a geometrical consequence of the above-mentioned conditions (the movements being supposed uniform),that the radius of each epicycle must at all times be parallelto a line joining the Earth and the Sun , as is shown inFig . XXX. ; but we are not quite certain how far this lastfact was clearly understood; nor are we sure that the exactyearly period of Mars in its epicycle was fully realized. For,if all this had been duly appreciated, we should suppose thata simple application of such geometrical principles as themathematicians of those days were perfectly acquainted with,would have led them to argue as follows :—The movementof each planet in its epicycle being the same, such movementis probably only apparent, and caused by a real movement ofthe Earth , according to the law, that, if an observer be situatedupon a moving body, the apparent motion produced in any-thing which he looks at, will be just the same as if he werebrought to rest, and a velocity, the reverse of his own, werecommunicated to the other body.
We have stated that each of the exterior planets was neces-sarily supposed to describe its epicycle in one year; nevertheless,it should be noticed that Jupiter and Saturn would occupysomewhat longer than a year in describing complete loopsof their respective apparent orbits (such as we showed inFig . XXV.); because the onward march of the centre of theepicycle would prevent the radius joining it to the planet, frompointing again to the Earth at the end of one epicyclic revolu-tion. Before it could do so the radius would need to turnround through an additional angle equal to that which thecentre had in the meantime described round the Earth . Thismay be easily seen in the subjoined view of one of the loops