128 A SHORT HISTORY OF SCIENCE 



The earth is a sphere, situated in the centre of the heavens; ii 

 it were not, one side of the heavens would appear nearer to us than 

 the other, and the stars would be larger there ; if it were on the celes- 

 tial axis but nearer to one pole, the horizon would not bisect the equator 

 but one of its parallel circles; if the earth were outside the axis, 

 the ecliptic would be divided unequally by the horizon. The earth 

 is but as a point in comparison to the heavens, because the stars appear 

 of the same magnitude and at the same distances inter se, no matter 

 where the observer goes on the earth. It has no motion of translation, 

 first, because there must be some fixed point to which the motions of the 

 others may be referred, secondly, because heavy bodies descend to the 

 centre of the heavens which is the centre of the earth. And if there was 

 a motion, it would be proportionate to the great mass of the earth and 

 would leave behind animals and objects thrown into the air. This 

 also disproves the suggestion made by some, that the earth, while 

 immovable in space, turns round its own axis, which Ptolemy ac- 

 knowledges would simplify matters very much. 1 



Chapter IX explains the calculation of a table of chords. Start- 

 ing with the chords of 60 and 72, already known as sides of regular 

 polygons, he devises ingenious geometrical methods for finding 

 chords of differences and of half-angles. Thus he computes the 

 chords for 12, 6, 3, l, and f. Hipparchus had already com- 

 puted such a table, but Ptolemy completes it by showing that 



f chord 1| < chord 1 < $ chord f 



and thence deriving close approximations for the chords of 1 

 and ^ and constructing a table for each half-degree up to 180. 

 His results are expressed in sexagesimal fractions of the radius (of 

 which they are thus numerically independent) and are equivalent 

 in accuracy to five decimals in our notation. He also employs 

 our present method of interpolation skilfully. This chapter is 

 the culmination of Greek trigonometry, which owed its further 

 development to Indian and Arabic mathematicians. 



1 "For Ptolemy more geometer and astronomer than philosopher, the astronomer 

 who seeks hypotheses adapted to save the apparent movements of the stars knows 

 no other guide than the rule of greatest simplicity : It is necessary as far as possible 

 to apply the simplest hypotheses to the celestial movements, but if they do not 

 suffice, it is necessary to take others which fit better. " Duhem. 



