A history of Greek mathematics - Wilbourhall.org

;
Hence N :
ARISTARCHUS OF SAMOS 13
LC = ON 2 : NL
2
> 89 2 :45 2 ;
therefore ON: LP > 7921 :
4050
> 88 : 45, says Aristarchus.
[If Jffjy be developed as a continued fraction, we easily
i ,
obtain 1 H , which is in tact
1+21+2 45 J
• 1 ! 1 i • t • • o ,
88 ~]
(2) ON < 2 (diam. of moon).
But (diam. of moon) < ^g (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 BUV (Fig. 1) and the triangle BLN (Fig. 2) are
similar, and
But . LN
LN:LP = UV: (diam. of sun)
= WU:UA
= UA:AS
:
therefore, a fortiori, UA :
And
LP
< UA:AY.
> 89 : 90
AY > 89 : 90.
UA:AY=2UA:YZ
= (diam. of sun) : YZ.
(Prop. 12)
But ON: (diam. of sun) > 22 : 225
;
(Prop. 13)
therefore, ex aequali,
ON: YZ > 89 x 22 : 90 x 225
> 979:10125.