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 observation, 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.[1]
- ↑ 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 θ + b sin (2Φ — θ), where a, b are two numerical quantities, in round numbers 6° and 1°, θ is the angular distance of the moon from perigee, and Φ is the angular distance from the sun. At conjunction and opposition Φ is 0° or 180°, and the two terms reduce to (a—b) sin θ. 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 θ + sin (2 Φ — θ)], or 2 b sin Φ cos (Φ — θ),
which vanishes at conjunction or opposition, but reduces at the quadratures to 2 b sin θ, which again vanishes if the moon is at apogee or perigee (θ = 0° or 180°), but has its greatest value half-way between, when θ = 90°. Ptolemy's construction gave rise also to a still smaller term of the type,
c sin 2 Φ [cos (2 Φ + θ) + 2 cos (2 Φ — θ)],
which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.
