Note 32 — Chapter II · GREEK ASTRONOMY.

The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude, a sinθ + 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) sinθ. 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 [sinθ + sin(2φ - θ)], or 2 b sinφ cos (φ-θ),

which vanishes at conjunction or opposition, but reduces at the quadratures to 2 b sinθ, 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 2φ [cos (2φ + θ) + 2 cos (2φ - θ)],

which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.