I thank Julio for citing Aritotle's text on Physics.
Seen from our modern base 10 decimal point of view,
infinite series are commonly assumed to be the best
even though simple numbers like 1/3 must be rounded off,
seen as:
1/3 = 3/10 + 3/10^2 + 3/10^3 + ... + 2/10^n + ...
at some point.
Aristotle, Plato, Pythagoras, Eudoxus, Archimedes and
all Classical Greeks knew of such infinite series,
such as:
1/3 = 1/4 + 1/16 + 1/64 + 1/256 + ... + 1/(2^2n) + ...
however, for most problem solving situations, includes
physics problems (of Aristotle) exactness seemed to
be a requirement for Greeks, from time to time, so Greeks
expressed 1/3 = 3', or when needed:
1/3 = 1/4 + 1/(3*4)
as Archimedes is cited by Dijksterhuis, page 129, in the
book Archimedes:
"Given a series of magnitudes, each of which is equal to
four times the next in order, all the magnitudes and
one-third of the least added together will exceed the
greatest by one-third."
Dijkershuis used the statement:
4A/3 = A + B + C + D + E + E/3
where A could be any number, even pi, to represent
Archimedes' finite statement. However, looking for the
shortest series,
4A/3 = A + A/4 + A/(3*4)
is obviously much preferred over the infinite form:
4A/3 = A + A/4 + A/16 + A/64 + ...
as Babylonians any many modern physicists would view
the related arithmetic. Greeks knew better.
In conclusion, I have intended to show that the majority,
if not all, classical Greeks were trained in the original
Academy (as Pythagoras would have also known) as indicated
above. That is, Aristotle would have also known and selected
from finite and infinite versions of arithmetic situations,
sometimes chosing the finite when an exact answer was desired,
or an infinite series when no answer seemed available at the
time.
Any confusion relate to use of the terms infinite and finite
is only a modern concern. Ancient Greeks would have under-
stood the related arithmetic, algebra and geometry of the
situations, and therefore they would not had introduced
philosophical concerns to the degree that Aristotle is
usually translated.
Regards to all,
Milo Gardner