A history of Greek mathematics - Wilbourhall.org

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ARISTARCHUS OF SAMOS
Now BH:HA[ = sin HAB : sin HBA]
> lHAB.lHBA,
whence LEAB< &LHBA,
F D
and (taking the doubles) Z HAK < £ Z HBK.
But Z imST = Z #£(7 = ^o R (where E is a right angle)
therefore Z.HAK (arc CGD) :
;
90.
(arc EGF) = 89 : 90.
(arc EGF)
[equivalent to sin oc /sin ft > oc/ft; where /.CBD = 2 a,
and 2 /3 = 7r] ;
therefore CD.EF [= cos 1°] > 89 : 90,
while obviously CD :
EF
< 1.
Prop. 11 finds limits to the ratio EF:BA (the ratio of the
diameter of the moon to the distance of its centre from
the centre of the earth) ; the ratio is < 2 : 45 but > 1 : 30.
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