What are the Egyptian traditions that
Archimedes used to build his proofs?
The primary one that I would suggest
is the Middle Kingdom hieratic finite
series numeration tradition of double
checking Old Kingdom hieroglyphic
infinite series results.
As clearly known to Greeks, the Old
Kingdom Horus-Eye infinite series:
1) 1 = 1/2 + 1/4 + ... + 1/2n + ...
can be parsed into two geometric series:
2) 1/3 = 1/4 + 1/16 + 1/2n + ...
and,
3) 2/3 = 1/2 + 1/8 + 1/(2*4n) + ...
and exactly re-written as a finite series,
as the MMP, RMP, EMLR, Hibeh P. and other
Greek era documents detail.
Archimedes plausibly began with the traditional
infinite series in his famous 1/4th geometric series
and re-wrote it as a finite series as Dijksterhuis
cites as a proposition:
"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 greast by one-third"
as algebraically summarized as follows:
4) 4/3 = A + B + C + D + E + E/3
One proof aspect of Archimedes' final exact series
is simply a restatement of the 1/4th geometic infinite
series to a finite series, a method that historians
may agree is present in the RMP and its 2/nth table.
I further suggest that Archimedes followed a related
'Egyptian' method to prove his In Quadrature of the
Parabola proposition by:
let A = 1, then,
1/3 = 1/4 + 1/16 + 1/64 + 1/256 + ...
(exposing equation 2))
with a last introduced at any point,
such as E/3 by:
E/3 = 1/3 - 85/255 = (256 - 255)/(3*256)
Following a finite series method very much as Hultsch
suggested in 1895 was the basis for all RMP 2/p series.
All that I add to the above is that Archimedes
may have recursively computed his initial
geometric series by beginning with 1/3 then:
1/3 - 1/4 = (4 - 3)/12 = 1/(3*4)
1/12 - 1/16 = (16 - 12)/(12*16) = 1/(3*16)
1/48 - 1/64 = (64 - 48)/(48*64) = 1/(3*64)
1/192 - 1/256 = (256 - 192)/(192*256) = 1/(3*256)
with the last exact term E/3 being a conventional
five-term stopping point, one less than Old Kingdom
Egyptians had established for 6-term (rounded off)
Horus-Eye infinite series.
Comments?
Milo Gardner
Sacramento, Calif.