122 A SHORT HISTORY OF SCIENCE 



mation to the motions of the planets, and showed the necessity 

 both of a better theory and of more complete observational data. 

 It is interesting to speculate on the consequences which might have 

 resulted for astronomical science had the genius of Hipparchus 

 adopted the daring heliocentric theories of Aristarchus instead of 

 adhering to the traditional geocentric ideas. 



INVENTION OF TRIGONOMETRY. Not least important among the 

 services of Hipparchus to science was his laying the foundations 

 of trigonometry, by constructing for astronomical use a table of 

 chords, equivalent to our tables of natural sines. He gave also a 

 method for solving spherical triangles. It is said that he first 

 indicated position on the earth by latitude and longitude the 

 germ of coordinate geometry Eratosthenes having merely given 

 the latitude by means of the height of the pole-star. For mapping 

 the sky he used stereographic projection, for mapping the earth 

 orthographic. 



To sum up the chief work of Hipparchus : he made very effec- 

 tive use of extant records of earlier astronomers with critical con- 

 sideration of their value; he made a prolonged and systematic 

 series of observations with the best available instruments; he 

 worked out a consistent mathematical theory of the motions of the 

 heavenly bodies so far as his data warranted ; he made a new 

 catalogue of 1080 stars, with the classification by magnitude still in 

 use ; he discovered the precession of the equinoxes ; he laid the 

 foundations of trigonometry. 



Delambre, the great French historian of astronomy, says : 



When we consider all that Hipparchus invented or perfected and 

 reflect upon the number of his works and the mass of calculations 

 which they imply, we must regard him as one of the most astonishing 

 men of antiquity, and as the greatest of all in the sciences which are not 

 purely speculative, and which require a combination of geometrical 

 knowledge with a knowledge of phenomena, to be observed only by 

 diligent attention and refined instruments. 



In spite of these brilliant achievements, the position of Apollonius 

 and Hipparchus had become relatively isolated under the prevalent 



