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Egyptian algebraic geometry
Posted:
Aug 26, 2010 2:50 PM


Dear Forum members:
The Moscow Mathematical Papyrus (MMP 10), the Kahun Papyrus (KP) and the Rhind Mathematical Papyrus (RMP 41, 42, 43) and ) used area and volumes of a circular granary by four algebraic formulas. The formulas were defined by radius R being replaced by onehalf a diameter (D/2), and pi replaced by 256/81 in the well known area of a circle formula. Volume formulas added height (H) and other algebraic geometry considerations:
MMP 10 reported:
http://moscowmathematicalpapyrus.blogspot.com/
1. A (cubit^2) = ([8/9)(D)(8/9)(D)] (cubit^2 not named)
with D = 9
the scribe wrote:
(9  9/9) = (8)(8/2) = 32 The KP reported algebraic geometry in
http://planetmath.org/encyclopedia/KahunPapyrusAndArithmeticProgressions.html
2. 4. V(Khar) = (2/3)(H)(4/3)(D)(4/3)(D)
with D = 12, H = 8
the scribe wrote:
V =(2/3)(8)= 16/3 times (4/3)(12)= (16)(16) = 4096/3
recorded (1365 1/3)Khar
RMP 41, 42 and 43 reported three algebraic geometry formulas in
http://ahmespapyrus.blogspot.com/2009/01/ahmespapyrusnewandold.html by
3. A (cubit^2) = (8/9)(D)(8/9)(D)
was used in RMP 41 let D = 9
and wrote:
(9  9/9) = (8)(8) =64 cubit^2
4. V(cubit^3) = (H)(8/9)(D)(8/9)(D)
was used in RMP 42 with D=10, H= 10 writing
a. (10  10/9) = (80/9)(80/9) = 6400 cubits^2
b. 64000 cubits^3
5. V (Khar) = (3/2)(H)(8/9)(D)(8/9)(D)
in RMP 42 64000 cubits^2 were scaled to Khar by halving 64000, finding 32000, and reporting 96000 khar also reporting hundred weights of hekats, and number of hekats.
6. V(Khar) = (2/3)(H)((4/3)(D)(4/3)(D)
was used in RMP 43 letting: D = 8, H = 6
with Ahmes writing:
(2/3)(6) = 4, and (4/3)(8) = (32/8)(32/8) = 4096/9 =
(455 + 1/9)khar
and the KP letting D = 12, H = 8
and the scribe writing
2/3(8) = 16/3; (4/3)(12) = (16)(16) = 4096/3 =
(1365 + 1/3)khar
Q.E.D.
Milo Gardner



