72 A Short History of Astronomy [Cn. it. 



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 re- 

 present 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 i4th and i5th centuries ought to have 

 objected (as some of them actually did), to this innova- 

 tion of Ptolemy's. But there seems no good reason fo* 

 attributing this a priori attitude to the later scientific Greek 

 astronomers (cf. also 38, 47).* 



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 

 vine 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 re- 

 spectively) 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 in- 

 troduced by Kepler. In elliptic motion the angular motion and 

 distance are represented approximately by the formulae nt + 2e sin nt 

 a (i cos nt} respectively; the corresponding formulae given by 

 the use of the simple eccentric are nt + e' sin nt, a (i 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 con 

 ditions are therefore inconsistent. But by the introduction of an 

 equant the formulae become nt + 2e' sin nt, a (l e' cos nf), 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. 



