Ei 
■ 
3. 
8 : 43 
; 19. 
are proved. 
LP. 
ARISTARCHUS OF SAMOS 13 
Hence ON: LC = ON 2 : NL 2 
> 89 2 : 45 2 ; 
therefore ON:LP > 7921 :4050 
> 88 : 45 ^ says Aristarchus. 
[if im be developed as a continued fraction, we easily 
obtain 1 + " \, which is in fact • J 
(by Prop. 12) 
— - j wnicn is in tact _ 
1+21+2 45 
(2) ON < 2 (diam, of moon). 
But (diam. of moon) < Ag (diam. of sun); (Prop. 7) 
therefore ON < ^ (diam. of sun). 
Again ON: (diam. of moon) > 88 : 45, from above, 
and (diam. of moon): (diam. of sun) > 1:20; (Prop. 7) 
therefore, ex aequali, 
ON: (diam. of sun) > 88:900 
> 22 : 225. 
(3) Since the same cone comprehends the sun and the moon, 
the triangle BJJV (Fig. 1) and the triangle BLN (Fig. 2) are 
similar, and 
LN-.LP = UV: (diam. of sun) 
= WU: UA 
= UA:AS 
< UA: AY, 
But LN:LP > 89:90; (Prop. 12) 
therefore, a fortiori, UA: AY > 89:90. 
And UA:AY =2UA:YZ 
= (diam. of sun): YZ. 
But ON: (diam. of sun) > 22:225; (Prop. 13) 
therefore, ex aequali, 
• ON: YZ > 89 X 22 : 90 x 225 
> 979 :10125.