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
early geometry 1 / early geometry 2
/ early geometry 3 / early geometry 4 / early geometry 5 / early geometry 6
/ early geometry 7 / early
geometry 8 / early geometry 9 / early geometry 10
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
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
quadruple 456-361-288-649 fingers
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
quadruple 456-361-288-649
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
early geometry 1 / early geometry 2
/ early geometry 3 / early geometry 4 / early geometry 5 / early geometry 6
/ early geometry 7 / early
geometry 8 / early geometry 9 / early geometry 10