)S 
ARISTARCHUS OF SAMOS 
found in Prop. 4, 
;ircle drawn with 
lat on which the 
move as thejnoon 
is the radius AC 
nds at the centre 
D subtends at A 
led by r) 
BA = FE.BA. 
3t interest on the 
ista/ace of the sun 
but less than 20 
arth. This result 
vious attempts to 
speculation on the 
11-545 B. C.), who 
un and moon from 
>xus, according to 
n 9 times that of 
ir, 12 times; and, 
he two bodies are 
the same. 
A is the centre of 
if the moon at the 
B is right. ABEF 
sun’s circular orbit, 
so that 
I. Now, by Hypothesis 4, ¿ABC = 87°, s 
so that Z HBE = Z BAG = 3°; 
therefore Z GBE: Z HBE — £ R : R 
= 15:2, 
so that GE: HE [ = tan GBE: tan HBE] > Z GBE: Z HBE 
> 15 : 2. (1) 
The ratio which has to be proved >18:1 is AB:BC or 
FE.EH. 
Now FG: GE = FB: BE, 
whence FG 2 : GE 2 = FB 2 : BE 2 = 2 :1, 
and FG : GE = \/2:l 
>7:5 
(this is the approximation to V2 mentioned by Plato and 
known to the Pythagoreans).