equant to represent an irregular motion, if he had found that the motion was thereby represented with accuracy. The criticism appears to me in fact to be an anachronism. The earlier Greeks, whose astronomy was speculative rather than scientific, and again many astronomers of the Middle Ages, felt that it was on a priori grounds necessary to represent the "perfection" of the heavenly motions by the most "perfect" or regular of geometrical schemes; so that it is highly probable that Pythagoras or Plato, or even Aristotle, would have objected, and certain that the astronomers of the 14th and 15th centuries ought to have objected (as some of them actually did), to this innovation of Ptolemy's. But there seems no good reason for attributing this a priori attitude to the later scientific Greek astronomers (cf. also §§ 38, 47).[1]
It will be noticed that nothing has been said as to the actual distances of the planets, and in fact the apparent motions are unaffected by any alteration in the scale on which deferent and epicycle are constructed, provided that both are altered proportionally. Ptolemy expressly states that he had no means of estimating numerically the distances of the planets, or even of knowing the order of the distance of the several planets. He followed tradition in accepting conjecturally rapidity of motion as a test of nearness, and placed Mars, Jupiter, Saturn (which perform the circuit of the celestial sphere in about 2, 12, and 29 years respectively) beyond the sun in that order. As Venus and
- ↑ 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 formulæ 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 c' = c; 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.
