Heinz's review of partial fraction was interesting. I would like to
go beyond the Euclidean algorithm, and all algorithms (noting that
algorithms did not appear until Arab mathematicians developed the
tool). Clearing converting p/q into a concise unit fraction series
was generally achieved by Greeks (and Egyptians) outside of gcd's,
and modern number theory methods.
Historically, it is easy to prove that Archimedes built calculus not
on partial fractions, but on Egyptian fractions, as the 1/4th geometric
series per his lemma, algebraically stated as:
4A A A
--- = A + ---- + -----
3 4 (3*4)
where A = any number, even pi, clearly states.
The Egyptian fraction foundation is seen by removing A, and moving one
term, as given by:
4 1 1
-- = 1 + --- + ------
3 4 (3*4)
and re-arranging to an Egyptian fractions form, an algebraic identity,
a form short of an algorithm, or:
1 1 1
--- = --- + ----
3 4 12
That is to say, whenever a rational number p/q could be converted
to a finite series, as Greeks and Egyptians generally achieved by
converting n/p, n/pq, ..., 2/p, 2/pq, 1/p and 1/pq, hints of
Archimedes's lemma can be seen, a slice of a geometric shape being
found. All that was needed was to find the area of all the slices of
all geometric shapes to be computed to find a foundation of calculus.
Note the Archimedes' Method of Exhaustion may be mis-named, as
Dijksterhuis strongly suggested in his book Archimedes. The power of
The Method lies in converting infinite series to finite series. The
exhaustion aspect only points to finding every last slice of area,
say of a circle. The use of pi, for A, reduces the need a modern
limit > 0 type of thinking. Archimedes' calculus need not be seen as
using such a limit.
Stated another way, a mod n geometric series may have been an
Archimedean tool, as taught by Eudoxus, as a generalize tool that
goes well beyond the 1/4th geometric series, as suggested by:
1 1 1
----- = --- + ------- , one Egyptian fraction form,
(n -1) n n*(n -1)
and,
nA A A
------- = A + ---- + ------ , a proposed form of a generalized
(n -1) n n*(n-1) Archimedean lemma, mod n
that, of course, can be written out as an infinite series, as
Greeks and Egyptian wrote between, as needed, in either direction.
Regards to all,
Milo Gardner