8 
ARISTARCHUS OF SAMOS 
The first part follows from the relation found in Prop. 4, 
namely BC:BA < 1:45, 
for EF = 2 BG. 
The second part requires the use of the circle drawn with 
centre A and radius AC. This circle is that on which the 
ends of the diameter of the ‘ dividing circle’ move as thejnoon 
moves in a circle about the earth. If r is the radius AC 
of this circle, a chord in it equal to r subtends at the centre 
A an angle of or 60°; and the arc CD subtends at A 
an angle of 2°. 
But (arc subtended by CD): (arc subtended by r) 
< CD: r, 
or 2 : 60 < CD:r ; 
that is, CD: GA > 1:30. 
And, by similar triangles, 
CL: GA = GB: BA, or CD :CA = 2 CB: BA = FE: BA. 
Therefore FE:BA > 1:30. 
The proposition which is of the greatest interest on the 
whole is Prop. 7, to the effect that the distance of the sun 
from the earth is greater than 18 times, but less than 20 
times, the distance of the moon from the earth. This result 
represents a great improvement on all previous attempts to 
estimate the relative distances. The first speculation on the 
subject was that of Anaximander {circa 611-545 e. c.), who 
seems to have made the distances of the sun and moon from 
the earth to be in the ratio 3:2. Eudoxus, according to 
Archimedes, made the diameter of the sun 9 times that of 
the moon, and Phidias, Archimedes’s father, 12 times; and, 
assuming that the angular diameters of the two bodies are 
equal, the ratio of their distances would be the same. 
Aristarchus’s proof is shortly as follows. A is the centre of 
the sun, B that of the earth, and G that of the moon at the 
moment of dichotomy, so that the angle ACB is right. ABEF 
is a square, and AE is a quadrant of the sun’s circular orbit. 
Join BE, and bisect the angle FBE by BG, so that 
Z QBE = \R or 22|°. 
I. Now, 
so that 
therefore 
K 
B 
so that GE. 
The ratio 
FE: EH. 
Now 
whence 
and 
(this is the 
known to tl