5 



A Short History of Astronomy 



[CH. II. 



it .with the known angular dia- 

 meters of the sun and moon, 

 he obtained, by a simple cal- 

 culation,* a relation between 

 the distances of the sun and 

 moon, which gives either when 



* In the figure, which is taken 

 from the De Revolutionibus of 

 Coppernicus (chapter iv., 85), 

 let D, K, M represent respectively 

 the centres of the sun, earth, and 

 moon, at the time of an eclipse of 

 the moon, and let s Q G, s R E denote 

 the boundaries of the shadow-cone 

 cast by the earth ; then Q R, drawn 

 at right angles to the axis of the 

 cone, is the breadth of the shadow 

 at the distance of the moon. We 

 have then at once from similar 

 triangles 



G K Q M : A D G K \\ M K : K D. 



Hence if K D = n . M K and .*. 

 also AD = n . (radius of moon), n 

 being 19 according to Aristarchus, 

 G K QM : n . (radius of moon) GK 



: ; i : 

 n . (radius oi moon) GK 



= n G K n QM 



.'. radius of moon + radius of 

 shadow 



= +~) (radius of earth). 

 By observation the angular radius 

 of the shadow was found to be 

 about 40' and that of the moon to 

 be 15', so that 



radius of shadow = f radius of moon; 

 .'. radius of moon 



= I'T C 1 + ~) (radius of earth). 

 But the angular radius of the moon 

 being 15', its distance is necessarily 

 about 220 times its radius, 



and .*. distance of the moon 



= 60 (i + -) (radius of the earth), 

 which is roughly Hipparchus's 

 result, if n be any fairly large 



FIG. 2O. The eclipse method 

 of connecting the distances 

 of the sun and moon. 



