Note 34 — Chapter II · GREEK ASTRONOMY. (continued 2)

The advantage derived from the use of the equant can be made clearer by a mathematical comparison with the elliptic motion introduced by Kepler. In elliptic motion the angular motion and distance are represented approximately by the formulae nt + 2e sin nt, a (1 - e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric are nt + e′ sin nt, a (1 - e′ cos nt). To make the angular motions agree we must therefore take e′ = 2e, but to make the distances agree we must take e′ = e; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ become nt + 2e′ sin nt, a (1 - e′ cos nt), and both agree if we take e′ = e. Ptolemy’s lunar theory could have been nearly freed from the serious difficulty already noticed (§ 48) if he had used an equant to represent the chief inequality of the moon; and his planetary theory would have been made accurate to the first order of small quantities by the use of an equant both for the deferent and the epicycle.