46 Lessons in Early Geometry, part 3/10 / provisional version in freestyle English / a corrected version will follow in March, April or May (hopefully) / Franz Gnaedinger / February 2003 / www.seshat.ch

Lesson 13

The Rhind Mathematical Papyrus was written by one Ahmose or Ahmes in around 1650 BC, as a copy of a lost scroll from around 1850 BC.

In problem no. 32, Ahmes carries out the following division:

2 divided by 1 1/3 1/4 equals 1 1/6 1/12 1/114 1/228

I prefer a simpler notation of unit fractions that comes

close to the hieratic notation:

2  divided by  1 '3 '4  equals  1 '6 '12 '114 '228

Let us play with these numbers and consider them as measurements of a right parallelepiped:

height  2 units

length  1 '3 '4 units

width   1 '6 '12 '114 '228 units

How long are the cubic diagonals? Simply

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           '4             '76     units

The royal cubit of the New Kingdom measured 52.5 centimeters. 1 royal cubit equals 7 palms or 28 fingers; 1 palm equals 4 fingers. If the unit of the above parallelepiped measures 4 royal cubits 2 palm, we obtain these measurements:

height........ 16 royal cubits 2 palms

length........ 12 royal cubits 6 palms 1 finger

width......... 10 royal cubits 2 palms

cubic diagonal 23 royal cubits 1 palm 1 finger

The same numbers can be found by means of a number column for the calculation of the square:

7     5    14

12 -- 19 -- 24

31    43    62

74   105   148

179   253   358

432   611   ...

12  19  24

19x24=456  19x19=361  12x24=288  361+288=649

24/12 = 2

19/12 = 1 '3 '4

24/19 = 1 '6 '12 '114 '228

2 :  1 '3 '4  =  1 '6 '12 '114 '228

Lesson 14

Divide 2 by any number a and you obtain b:

2 : a = b

Use these numbers as measurements of a right parallelepiped. It will be a magic parallelepiped with the following properties:

height            2 units

length or width   a units

width or length   b units

area base / top   ab square units

volume            2ab cubic units

cubic diagonal    a+b units

Let us imagine a granary in the shape of a magic parallelepiped: volume 500 cubic cubits, capacity 750 khars or 15,000 hekat, inner height 2 units or 10 cubits or 70 palms or 280 fingers, length x width 2 square units or 50 square cubits or 2,450 square palms or 39,200 square fingers.

Here are four exact solutions, one of them provided by Ahmes' equation 1 '2 '4 times 5 '2 '7 '14 equals 10 in problem no. 34 of the Rhind Mathematical Papyrus:

inner height     10 royal cubits

inner length     10 royal cubits

inner width       5 royal cubits

cubic diagonal   15 royal cubits

inner height    280 fingers (10 royal cubits or 2 units)

inner length    245 fingers (1 '2 '4 units)

inner width     160 fingers (5 '2 '7 '14 royal cubits)

cubic diagonal  405 fingers (RMP 34)

inner height    280 fingers

inner length    224 fingers

inner width     175 fingers

cubic diagonal  399 fingers

inner height    280 fingers or 70 palms

inner length    200 fingers or 50 palms

inner width     196 fingers or 49 palms

cubic diagonal  396 fingers or 99 palms

If height, length and width measure 70, 50 and 49 palms, the cubic diagonal measures exactly 99 palms. Now consider a very good approximation:

inner height    280 fingers

inner length    198 fingers

inner width     198 fingers

cubic diagonal  practically 396 fingers

Such a granary can be measured simply using a rope:

height          length       width

o-----------------o------------o------------o

10 royal cubits  198 fingers   198 fingers

o-----------------o-------------------------o

diagonal base        cubic diagonal