A history of Greek mathematics - Wilbourhall.org

ON PLANE EQUILIBRIUMS, II 81
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*.(2PN+BO):PN 2 .(2BO + PN).
The geometrical proof is somewhat difficult, and uses a very
remarkable Lemma which forms Prop. 9. If a, b, c, d, x, y are
straight lines satisfying the conditions
a b c , \
T = - = -j (a > o > c > a),
b c a v '
d
x
and
then must
a — d i( a ~ c )
2a + 4:b + 6c + 3d y
5a+ 106 + 10c + 5(i a — c
x + y = -fa.
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
Arenarhis).
I have already described in a previous chapter the remarkable
system, explained in this treatise and in a lost work,
'Apxai, 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
it would take to fill
to express the number of grains of sand which
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 document
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 yjo^h
of the circle of the zodiac or half a degree. Since Archimedes,
in order to calculate a safe figure
1523.2 Q
(not too small) for the size