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Re: Ariadne's thread
Posted:
Jan 17, 2012 2:19 AM
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RMP 44 and 45
These problems concern a container in the form of a cube
measuring 10 by 10 by 10 royal cubits. If you wish to
build such a granary you have to know the lengths of the
floor diagonals, of the face diagonals, and of the cubic
diagonals.
1 1 2
2 3 4
5 7 10
12 17 24
29 41 58
70 99 140
and so on
If the edge of a cube measures 10 royal cubits or 70 palms,
the diagonal of a face measures practically 99 palms
(mistake 0.4 millimeters).
1 1 3
2 4 6
1 2 3
3 5 9
8 14 24
4 7 12
11 19 33
30 52 90
15 26 45
41 71 123
112 194 336
56 97 168
and so on
If the edge of a cube measures 10 royal cubits or 70 palms
or 280 fingers = 5 x 56 fingers, the cubic diagonal measures
practically 5 x 97 = 485 fingers (mistake 0.5 millimeters).
Now we may go a step further and imagine a cone, a sphere,
a hemi-ellipsoid and a cylinder in the frame of the cube.
By using the simple value '50 of 157 for pi we obtain:
CONE
diameter base 10 royal cubits
area base 78 '2 square cubits
height 10 royal cubits
volume 261 "3 cubic cubits
SPHERE
diameter 10 royal cubits
volume 523 '3 cubic cubits
HEMI-ELLIPSOID
diameter base 10 royal cubits
area base 78 '2 square cubits
height 10 royal cubits
volume 523 '3 cubic cubits
CYLINDER
diameter 10 royal cubits
area base 78 '2 square cubits
height 10 royal cubits
volume 785 cubic cubits
Volumes CONE : HEMI-ELLIPSOID : CYLINDER = 1 : 2 : 3
The granary in the form of a cube has a volume of 1,000
cubic cubits, while a granary in the shape of a hemi-
ellipsoid in the frame of the same cube has a volume of
about 523 cubic cubits. When we transform the cube into
a right parallelepiped, the ratio of the volume of the
parallelepiped to the volume of the inscribed hemi-
ellipsoid remains the same, about 2. This generates
a very simple practical formula:
If you see a granary in the shape of a hemi-ellipsoid
and wish to estimate its volume, carry out the following
calculation: '2 x diameter base x diameter base x height
and you obtain the approximate volume of the granary.
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