In problem no. 32 of the Rhind Mathematical Papyrus, Ahmes divides
2 by 1 1/3 1/14 and obtains 1 1/6 1/12 1/114 1/228:
By following Ahmes, young pupils learn how to handle unit fractions
and unit fraction series, whereas advanced learners may solve a more
demanding problem. Let the edges of a right parallel-epiped measure
How long are the cubic diagonals?
Simply
Divide 2 by any number a and you obtain b:
Using these numbers you can define a 'magic' parallelepiped of the
following properties:
Let the capacity of a granary measure 500 cube cubits and find solutions
of the above type. All granaries have a height of 10 royal cubits, while
length and width can vary. Here is one of many solutions:
Allow a tiny mistake and you obtain this right-parallelepiped:
This granary can easily be measured out using a simple rope with knots:
(My interpretations of some 65 problems from the Rhind Mathematical Papyrus
are found on my web site www.seshat.ch)
Franz Gnaedinger Zurich
2 divided by 1 '3 '4 equals 1 '6 '12 '114 '228
height 2 units
length 1 '3 '4 units
breadth 1 '6 '12 '114 '228 units
1 '3 '4 plus 1 '6 '12 '114 '228 units
or
1 1 plus '3 '6 plus '4 '12 plus '114 '228 units
or
2 '2 '3 '76 units
2 : a = b
height 2 units
length or breadth a units
breadth or length b units
area base or top ab square units
volume 2ab cube units
cubic diagonal a+b units
inner height 10 royal cubits or 70 palms
inner length 50 palms
inner width 49 palms
cubic diagonal exactly 99 palms
(10 royal cubits = 70 palms = 280 fingers)
rp 280 by 198 by 198 fingers
cubic diagonal practically 396 fingers or 99 palms
10 royal cubits 198 fingers 198 fingers
o------------------------o-----------------o-----------------o
inner height inner length inner width
o------------------------o-----------------------------------o
diagonal base or top cubic diagonal