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Re: Ariadne's thread
Posted:
Jan 9, 2012 4:56 AM
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RMP 43
Ahmes calculates the capacity of a cylindrical granary
whose inner diameter measures 6 royal cubits while the
inner height measures 9 royal cubits. However, his result
is wrong, because in the middle of his calculation he
jumps from one formula to a different one. Perhaps an
intentional mistake to challenge his pupils? Here I am
not concerned with any mistake or possible intentions,
but only with the numbers of the cylinder.
If a circle of the diameter 9 rc and a square of the side
8 rc have the same area, the volume of the above cylinder
measures 256 cubic cubits. Now let me compare two cylinders:
diameter 6 or 9 royal cubits
height 9 or 6 royal cubits
If you remember my interpretation of RMP 41, you can
easily calculate the measurements of the second cylinder:
its round wall has the same area while its volume equals
'6 x 9 x 256 cubic cubits = 384 cubic cubits
Now please imagine a granary in the shape of a hemi-
ellipsoid within the frame of the second cylinder:
what is the volume?
The circle of the second cylinder has a diameter of 9
royal cubits. Let us first imagine a sphere with the same
diameter, 9 rc, and calculate its volume by means of the
formula
volume sphere = '6 diameter x diameter x diameter x pi
By using the value '81 of 256 or '81 x 256 for pi we obtain:
'6 x 9 x 9 x 9 x '81 x 256 ccc = 384 cubic cubits
What a tidy result: a cylinder with diameter 9 and height
6 and a sphere with diameter 9 have exactly the same volume.
Now for the ellipsoid. This geometric form is nothing
other than a sphere lengthened in one dimension. The
diameter of the above sphere measures 9 cubits in every
dimension while the vertical diameter of the ellipsoid
measures 6 + 6 = 12 cubits. We obtain the volume of
the ellipsoid by multiplying that of the sphere by a factor
of '9 x 12 = '3 x 4 as follows:
sphere: diameter 9 x 9 x 9 volume 384 ccc
ellipsoid: diameter 9 x 9 x 12 volume 512 ccc
Now the volume of the hemi-ellipsoid equals '2 x 512 ccc
= 256 cubic cubits: exactly the volume of the first cylinder.
A granary in the shape of a cylinder has a diameter of
6 royal cubits and a height of 9 royal cubits -- another
granary in the shape of a hemi-ellipsoid has a diameter
of 9 royal cubits and a height of 6 royal cubits --
the two granaries have exactly the same volume
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