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Topic: Ariadne's thread
Replies: 86   Last Post: Jan 28, 2013 5:19 AM

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Franz Gnaedinger

Posts: 330
Registered: 4/30/07
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


Date Subject Author
11/17/11
Read Ariadne's thread
Franz Gnaedinger
11/17/11
Read Re: Ariadne's thread
Milo Gardner
11/18/11
Read Re: Ariadne's thread
Franz Gnaedinger
11/18/11
Read Re: Ariadne's thread
Milo Gardner
11/19/11
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Franz Gnaedinger
11/19/11
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Milo Gardner
11/20/11
Read Re: Ariadne's thread
Franz Gnaedinger
11/20/11
Read Re: Ariadne's thread
Milo Gardner
11/20/11
Read Re: Ariadne's thread
Milo Gardner
11/21/11
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Franz Gnaedinger
11/22/11
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Franz Gnaedinger
11/22/11
Read Re: Ariadne's thread
Milo Gardner
11/23/11
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Franz Gnaedinger
11/24/11
Read Re: Ariadne's thread
Franz Gnaedinger
11/24/11
Read Re: Ariadne's thread
Franz Gnaedinger
11/24/11
Read Re: Ariadne's thread
Franz Gnaedinger
11/24/11
Read Re: Ariadne's thread
Milo Gardner
11/25/11
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Franz Gnaedinger
11/26/11
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Franz Gnaedinger
12/2/11
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Franz Gnaedinger
12/2/11
Read Re: Ariadne's thread
Milo Gardner
12/3/11
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Franz Gnaedinger
12/4/11
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Franz Gnaedinger
12/4/11
Read Re: Ariadne's thread
Milo Gardner
12/5/11
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Franz Gnaedinger
12/5/11
Read Re: Ariadne's thread
Milo Gardner
12/7/11
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Franz Gnaedinger
12/8/11
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Milo Gardner
12/10/11
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Franz Gnaedinger
12/12/11
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Franz Gnaedinger
12/12/11
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Milo Gardner
12/13/11
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Franz Gnaedinger
12/13/11
Read Re: Ariadne's thread
Milo Gardner
12/15/11
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Franz Gnaedinger
12/15/11
Read Re: Ariadne's thread
Milo Gardner
12/15/11
Read Re: Ariadne's thread
Milo Gardner
12/16/11
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Franz Gnaedinger
12/16/11
Read Re: Ariadne's thread
Milo Gardner
12/18/11
Read Re: Ariadne's thread
Franz Gnaedinger
12/18/11
Read Re: Ariadne's thread
Milo Gardner
12/19/11
Read Re: Ariadne's thread
Franz Gnaedinger
12/20/11
Read Re: Ariadne's thread
Franz Gnaedinger
12/20/11
Read Re: Ariadne's thread
Milo Gardner
12/21/11
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Franz Gnaedinger
12/22/11
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Franz Gnaedinger
12/23/11
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Franz Gnaedinger
12/24/11
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Franz Gnaedinger
12/29/11
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Franz Gnaedinger
1/2/12
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Franz Gnaedinger
1/3/12
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Milo Gardner
1/4/12
Read Re: Ariadne's thread
Franz Gnaedinger
11/28/11
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Velev, Petyr
1/6/12
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Franz Gnaedinger
1/6/12
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Milo Gardner
1/9/12
Read Re: Ariadne's thread
Franz Gnaedinger
1/17/12
Read Re: Ariadne's thread
Franz Gnaedinger
1/19/12
Read Re: Ariadne's thread
Franz Gnaedinger
1/19/12
Read Re: Ariadne's thread
Milo Gardner
1/27/12
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Franz Gnaedinger
2/10/12
Read Re: Ariadne's thread
Franz Gnaedinger
2/28/12
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Franz Gnaedinger
3/2/12
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Franz Gnaedinger
3/23/12
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Franz Gnaedinger
3/24/12
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Milo Gardner
4/9/12
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Franz Gnaedinger
4/10/12
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Franz Gnaedinger
4/13/12
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Franz Gnaedinger
4/17/12
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Franz Gnaedinger
4/18/12
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Franz Gnaedinger
4/18/12
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Franz Gnaedinger
5/5/12
Read Re: Ariadne's thread
Franz Gnaedinger
5/7/12
Read Re: Ariadne's thread
Franz Gnaedinger
5/7/12
Read Re: Ariadne's thread
Milo Gardner
5/8/12
Read Re: Ariadne's thread
Franz Gnaedinger
5/8/12
Read Re: Ariadne's thread
Milo Gardner
5/8/12
Read Re: Ariadne's thread
Franz Gnaedinger
5/8/12
Read Re: Ariadne's thread
Franz Gnaedinger
5/9/12
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Franz Gnaedinger
5/10/12
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Franz Gnaedinger
8/14/12
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Franz Gnaedinger
1/13/13
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Franz Gnaedinger
1/19/13
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Franz Gnaedinger
1/23/13
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Franz Gnaedinger
1/23/13
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Franz Gnaedinger
1/24/13
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Franz Gnaedinger
1/26/13
Read Re: Ariadne's thread
Franz Gnaedinger
1/28/13
Read Re: Ariadne's thread
Franz Gnaedinger

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