
Re: Ariadne's thread
Posted:
Jan 17, 2012 2:19 AM


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 hemiellipsoid 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
HEMIELLIPSOID 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 : HEMIELLIPSOID : 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 hemiellipsoid 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.

