ARISTARCHUS OF SAMOS 
15 
But AB :BC < 20:1; 
therefore, ex acquali, 
AB:BR < 13500 : 674 
< 6750:337, 
whence, by inversion and componendo, 
BA : AB > 7087 : 6750. (1) 
But AX:XR=YZ.N0 
< 10125:979; (Prop. 13) 
therefore, convertendo, 
XA : AR > 10125:9146. 
From this and (1) we have, ex acquali, 
XA :AB > 10125 X 7087:9146 x 6750 
> 71755875 : 61735500 
> 43:37, a fortiori. 
[It is difficult not to see in 43:37 the expression 1 + , 
6+6 
which suggests that 43:37 was obtained by developing the 
ratio as a continued fraction.] 
Therefore, convertendo, 
XA :XB < 43:6, 
whence (diam. of sun) : (diam. of earth) <43:6. Q. E. D.