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Re: Ariadne's thread
Posted:
Dec 18, 2011 4:40 AM
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I didn't study the Egyptian Mathematical Leather Roll
and can only make a guess about the line
'8 equals '25 '15 '75 '200
It may stem from a multiple column of pi fractions:
1 1 1 '8 ('60) --- pi
'2 '2 '2 '16 ('120) --- '2 of pi
'3 '3 '3 '24 ('180) --- '3 of pi
'4 '4 '4 '32 ('240) --- '4 of pi
'5 '5 '5 '40 ('300) --- '5 of pi
'6 '6 '6 '48 ('360) --- '6 of pi
...
'5 '5 '5 '40 equals '5 '3 '15 '40
'5 '3 '15 '40 ('300) --- '5 of pi
'10 '6 '30 '80 ('600) --- '10 of pi
¹15 '9 '45 '120 ('900) --- '15 of pi
'20 '12 '60 '160 ('1200) --- '20 of pi
'25 '15 '75 '200 ('1500) --- '25 of pi or '8 ('1500)
'30 '18 '90 '240 ('1800) --- '30 of pi
...
The pi values are 25/8 and 377/120, from the sequence
3/1 (plus 22/7) 25/8 ... 377/120
For the time being I stay with the Rhind Mathematical
Papyrus. RMP32 revealed the 'magic parallel-epiped'
2 by A by B of the volume 4 and the diagonal A plus B.
How about the cube and its diagonal? Can it be defined
as easily? No, the diagonals of the square and cube
can only be approximated, though in a systematic way,
as shown by the next problem, again on the advanced
level.
RMP 33 - a wooden container in the shape of a cube
37 divided by 1 "3 '2 '7 equals 16 '56 '679 '776
Imagine a wooden chest in the shape of a cube. The chest
plus the lid measure 41 by 41 by 41 fingers. The boards
are 2 fingers strong. The inner space measures 37 by 37
by 37 fingers. How long are the diagonals of the outer
faces and of the outer cube in fingers? and of the inner
cube in palms?
Consult the number column approximating the square root
of 2. The first lines are 1 1 2 / 2 3 4 / 5 7 10 / 12 17 24 /
29 41 58 ...
If the side of a square measures 41 fingers, the diagonal
measures 58 fingers.
Now let us consult the number column for the square root
of 3. The first lines are 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 ...
The wooden container measures 41 by 41 by 41 fingers,
the diagonals of the faces measure 58 fingers, and
the diagonals of the outer cube measure 71 fingers.
The cavity measures 37 by 37 by 37 fingers. How long is
the cubic diagonal? Use the numbers 97 and 168. Multiply
37 fingers by '97 of 168 and you obtain the diagonal in
fingers. Divide 37 fingers by '42 of 97 = 1 "3 '2 '7
and you obtain the diagonal in palms. Ahmes found the
number 16 '56 '679 '776, hence the diagonal measures
practically 16 '56 '679 '776 palms or about 16 palms.
And the volume? A surprise: it measures practically
1 "3 '2 '7 cubic cubits.
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