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Topic: Egyptian algebraic geometry
Replies: 11   Last Post: Sep 22, 2010 9:00 AM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
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 one-half 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/ahmes-papyrus-new-and-old.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

Date Subject Author
8/26/10 Milo Gardner
8/27/10 Milo Gardner
8/30/10 Milo Gardner
9/7/10 Milo Gardner
9/7/10 Milo Gardner
9/8/10 Milo Gardner
9/10/10 Milo Gardner
9/13/10 Milo Gardner
9/16/10 Milo Gardner
9/18/10 Milo Gardner
9/20/10 Milo Gardner
9/22/10 Milo Gardner