82 
ARCHIMEDES 
of the universe, has to make certain assumptions as to the 
sizes and distances of the sun and moon and their relation 
to the size of the universe, he takes the opportunity of 
quoting earlier views. Some have tried, he says, to prove 
that the perimeter of the earth is about 300,000 stades; in 
order to be quite safe he will take it to be about ten times 
this, or 3,000,000 stades, and not greater. The diameter of 
the earth, like most earlier astronomers, he takes to be 
greater than that of the moon but less than that of the sun. 
Eudoxus, he says, declared the diameter of the sun to be nine 
times that of the moon, Phidias, his own father, twelve times, 
while Aristarchus tried to prove that it is greater than 18 but 
less than 20 times the diameter of the moon; he will again be 
on the safe side and take it to be 30 times, but not more. The 
position is rather more difficult as regards the ratio of the 
distance of the sun to the size of the universe. Here he seizes 
upon a dictum of Aristarchus that the sphere of the fixed 
stars is so great that the circle in which he supposes the earth 
to revolve (round the sun) ‘bears such a proportion to the 
distance of the fixed stars as the centre of the sphere bears to 
its surface ’. If this is taken in a strictly mathematical sense, 
it means that the sphere of the fixed stars is infinite in size, 
which would not suit Archimedes’s purpose ; to get another 
meaning out of it he presses the point that Aristarchus’s 
words cannot be taken quite literally because the centre, being 
without magnitude, cannot be in any ratio to any other mag 
nitude ; hence he suggests that a reasonable interpretation of 
the statement would be to suppose that, if we conceive a 
sphere with radius equal to the distance between the centre 
of the sun and the centre of the earth, then 
(diam. of earth): (diam. of said sphere) 
= (diam. of said sphere): (diam. of sphere of fixed stars). 
This is, of course, an arbitrary interpretation; Aristarchus 
presumably meant no such thing, but merely that the size of 
the earth is negligible in comparison with that of the sphere 
of the fixed stars. However, the solution of Archimedes’s 
problem demands some assumption of the kind, and, in making 
this assumption, he was no doubt aware that he was taking 
a liberty with Aristarchus for the sake of giving his hypo 
thesis an air of authority. 
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