66 A Short History of Astronomy [On. II, 



means of an eccentric, and depending only on the position 

 of the moon with respect to its apogee. Ptolemy, however, 

 discovered, what Hipparchus only suspected, that there 

 was a further inequality in the moon's motion to which 

 the name evection was afterwards given and that this 

 depended partly on its position with respect to the sun. 

 Ptolemy compared the observed positions of the moon with 

 those calculated by Hipparchus in various positions relative 

 to the sun and apogee, and found that, although there was 

 a satisfactory agreement at new and full moon, there was a 

 considerable error when the moon was half-full, provided 

 it was also not very near perigee or apogee. Hipparchus 

 based his theory of the moon chiefly on observations of 

 eclipses, i.e. on observations taken necessarily at full or new 

 moon ( 43), and Ptolemy's discovery is due to the fact 

 that he checked Hipparchus's theory by observations taken 

 at other times. To represent this new inequality, it was 

 found necessary to use an epicycle and a deferent, the latter 

 being itself a moving eccentric circle, the centre of which 

 revolved round the earth. To account, to some extent, for 

 certain remaining discrepancies between theory and obser- 

 vation, which occurred neither at new and full moon, nor 

 at the quadratures (half-moon), Ptolemy introduced further 

 a certain small to-and-fro oscillation of the epicycle, an 

 oscillation to which he gave the name of prosneusis.* 



* The equation of the centre and the evection may be expressed 

 trigonometrically by two terms in the expression for the moon's 

 longitude, a sin + b sin (2 0), where a, b are two numerical 

 quantities, in round numbers 6 and 1, 6 is the angular distance of 

 the moon from perigee, and is the angular distance from the sun. 

 At conjunction and opposition is O or 180, and the two terms 

 reduce to (a 6) sin 6. This would be the. form in which the 

 equation of the centre would have presented itself to Hipparchus. 

 Ptolemy's correction is therefore equivalent to adding on 



b [sin e + sin (2 0)], or 2 b sin cos (0-0), 



which vanishes at conjunction or opposition, but reduces at the 

 quadratures to 2 b sin 0, which again vanishes if the moon is at apogee 

 or perigee (0 = o or 180), but has its greatest value half-way 

 between, when = 90. Ptolemy's construction gave rise also to 

 a still smaller term of the type, 



c sin 2 [cos (204 0) + 2 cos (2 0)], 



which, it will be observed, vanishes at quadratures as well as at 

 conjunction and opposition. 



