454 Proceedings of the Royal Irish Academy. 



procediaxe "svill be to begin "witb chaps. 2 and 3 of Book tti., since it 

 is tbere that Tve have Ptolemy's account of '^hat Hipparchus Tvrote on 

 the subject. 



We find that, when Hipparchus mentions the equinoctial points, as 

 he does in the title of his tractate on the precession, and in the text of 

 another ti'actate quoted by Ptolemy, he speaks of them as shifting, 

 which, in this case, as we shall see, inTolves reti'ogressing. But when 

 he mentions the stars with which he is immediately engaged he speaks 

 of them as progi'essing, without adding "relatively" in either case. 

 This is natural enough for convenience ; but the great probability is 

 that some of the writers referred to above abide by one, and some by 

 the other, and that this is to a great extent the origin of the disagi^ee- 

 ment between them. If Hipparchus' own words, as handed down to 

 us by Ptolemy, be taken by themselves, they will not enable us to 

 settle the question now in hand, ^e must consider the treatment of 

 them by Ptolemy, who was in a better position than we ai'e for under- 

 standing them. 



Before proceeding, let us say a few words more on a paii:icular just 

 referred to. The title of Hipparchus' ti'actateis, " Concerning the shifting 

 of the tropical and equinoctial points." Hipparchus' word for ' shifting,' 

 /AeraTTTwcrts, could mean shifting forwards, as well as backwards. But, 

 in the present case, any change of position, except a backward one, is 

 altogether out of the question. If, then, there should be any shifting, 

 or change of position, of the equinoctial points on the ecliptic, it must 

 be a retrograde one ; and therefore Hipparchus' word, /xera-rcocris, 

 necessarily involves this meaning, though it does not etymologically 

 express it. "We can, therefore, with Delambre and others, represent, 

 though not translate, the title of Hipparchus' tractate thus — " Concern- 

 ing the retrogradation of the tropical and eq^imioctial points." 



But, as we have said, Hipparchus, in his discussion of the stars 

 more particulaiy in question, speaks of their progressive motion. 

 Although at first sight this may seem inconsistent, it is really not so. 

 When we are thinking simply of the increase of the longitudes of 

 the fixied stars, or the mere relative motion between them and the 

 eqtiinoctial points, it comes to the same whether we speak of the pro- 

 gression of the stars, or the reti'ogi-ession of the equinoctial points, 

 from one of which those longitudes are reckoned. Just as the relative 

 motion between the Sun and the eastern horizon, in the morning, which 

 we know to be due to the downward tiu-ning of the horizon there, is 

 almost always spoken of, even by the asti'onomers themselves, as the 

 rising of the Sun. This last expression is not to be taken literally. 



