ON PLANE EQUILIBRIUMS, II 
81 
Uolic segments, 
srs in the same 
segments only, 
ired of any two 
sntre of gravity 
bion. It is the 
iple equation in 
s' the centre of 
of the two seg- 
these segments, 
since the centres 
segments divide 
HH' is parallel 
f gravity of the 
tat K, the point 
rama 3 to Prop. 
L V = A V. But 
ame ratio as G 
lerefore 
) = m.AV. 
le triangle ABB' 
(dividing out by 
f gravity of the 
die! chords PF', 
ad the diameter 
actively, Archi- 
r e equal parts of 
to N than M is), 
the centre of gravity G of the portion of the parabola between 
PP' and BB' divides LM in such a way that 
LG-.GM = BO 2 . (2 PJy + BO): PN 2 . (2 BO + PUT). 
The geometrical proof is somewhat difficult, and uses a very 
remarkable Lemma which forms Prop. 9. If a, b, c, d, x, у are 
straight lines satisfying the conditions 
а Ъ c j \ 
1= c = d ia>h>C>d) • 
d _ x 
a — d f (a — c) у 
■. 2<x-( _ 46-)-6c-( _ 3cZ у 
and ; = s 
5a+ 106 + 10c + 5d a — c 
then must x + y = f a. j 
The proof is entirely geometrical, but amounts of course to 
the elimination of three quantities b, c, d from the above four 
equations. 
The Sand-reckoner (Psammites or Arenarius). 
I have already described in a previous chapter the remark 
able system, explained in this treatise and in a lost work, 
’Ap-fcai, Principles, addressed to Zeuxippus, for expressing very 
large numbers which were beyond the range of the ordinary 
Greek arithmetical notation. Archimedes showed that his 
system would enable any number to be expressed up to that 
which in our notation would require 80,000 million million 
ciphers and then proceeded to prove that this system more 
than sufficed to express the number of grains of sand which 
it would take to fill the universe, on a reasonable view (as it 
seemed to him) of the size to be attributed to the universe. 
Interesting as the book is for the course of the argument by 
which Archimedes establishes this, it is, in addition, a docu 
ment of the .first importance historically. It is here that we 
learn that Aristarchus put forward the Copernican theory of 
the universe, with the sun in the centre and the planets 
including the earth revolving round it, and that Aristarchus 
further discovered the angular diameter of the sun to be 7-totli 
of the circle of the zodiac or half a degree. Since Archimedes, 
in order to calculate a safe figure (not too small) for the size