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Re: Ariadne's thread
Posted:
Dec 22, 2011 3:15 AM
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RMP 37 - a cone
Ahmes finds a volume that measures '4 '32 hekat or 90 ro.
1 cubic cubit equals 30 hekat. 1 hekat equals 320 ro.
90 ro can be given as a cone with these measurements
(mistake '4 ro):
diameter base 7 fingers
circumference 22 fingers
height 4 palms
RMP 38 - transforming a square hekat into a cylinder
Let me begin with the subdivision of the royal cubit rod
of Amenhotep I:
1 royal cubit (52.5 cm) = 7 palms (7.5 cm) = 28 fingers
(1.875 cm) = 56 Re marks = 84 Shu marks = 112 Tefnut
marks = 140 Geb marks = 168 Nut marks = 196 Osiris marks
= 224 Isis marks = 252 Seth marks = 280 Nephtys marks =
308 Horus marks = 336 Imsety marks = 364 Hapy marks =
392 Duamutf marks = 420 Qhebsenuf marks = 468 Thoth marks
Now for RMP 38. Ahmes comes up with a funny equation:
1 hekat x 3 '7 x '22 x 7 = 1 hekat
The hekat is a measure of capacity. 30 hekat equal
1 cubic cubit. 1 hekat may be defined as a right
parallelepiped with the following measurements:
'2 royal cubit x '3 royal cubit x '5 royal cubit
or
28 Re marks x 28 Shu marks x 28 Geb marks
In Qhebsenuf marks:
210 Qm x 140 Qm x 84 Qm
cubic diagonal exactly 266 Qm
according to the quadruple 6-10-15-19
A hekat in the shape of a right parallelepiped is
well-defined. How about other shapes? Let us look again
at Ahmes' equation:
1 hekat x 3 '7 x '7 x 22 = 1 hekat
The numbers 3 '7 and '7 of 22 remind us of pi and 1/pi.
How about a hekat in the shape of a cylinder? I replace
the left hekat with the above definition:
210 Qm x 140 Qm x 84 Qm x 3 '7 x '22 x 7 = 1 hekat
Now I transform my equation:
'4 x 105 Qm x 105 Qm x 3 '7 x '11 x 3136 Qm = 1 hekat
The first term
'4 x 105 Qm x 105 Qm x 3 '7 or '4 x 7 f x 7 f x 3 '7
can be regarded as the area of a circle whose diameter
measures 105 Qhebsenuf marks or 7 fingers, while the
second term
'11 x 3136 Qm = 19 '165 fingers
can be regarded as the height of a cylinder. I leave out
the small fraction '165 and keep the height 19 fingers.
Now I can define my cylindrical hekat and quadruple-hekat
simply:
diameter 7 fingers / 14 fingers
circumference 22 fingers / 44 fingers
height 19 fingers / 19 fingers
volume 1 hekat / 4 hekat
The mistakes are tiny.
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