46 Lessons in
Early Geometry, part 9/10 / 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
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geometry 4 / early geometry 5 / early geometry 6 / early geometry 7
/ early geometry 8 / early geometry 9 / early geometry 10
Lesson 39
Gregory Peacock and Augustus de Morgan, the
founding fathers of symbolic algebra, freed the algebraic operations from the
restrictions of conventional numbers. A similar and no less radical step of
emancipation occurred several millennia ago, when the numbers were freed from
objects, and when counting passed on to calculating and computing (from Latin
computare, con-putare, think together, add up in the mind). As long as the
numbers belong to objects you can mainly count:
one fish
two fish three fish many fish
1 loaf of bread, 2 loaves, 3 loaves, 4
loaves, 5 loaves
But when you free the numbers from the objects
and see them as ideal objects in their own right you can perform all kinds of
new tasks. Consider for example this problem. 700 loaves of bread shall be
divided among four men called A B C D, in such a way that A obtains the double
share of C, and B obtains the double share of D, while A and C together obtain
the double share of B.
Let me try to solve this problem by playing
with beans of different color, here represented by the letters a b c d:
a a a a a a a a c c c c
b b b b b b d d d
8
beans 4 beans 6 beans
3 beans
I found a total of 8+4+6+3 = 21 beans, which
represent 700 loaves of bread.
A's share will be '21 of 8 times 700 loaves,
makes 266 "3 loaves.
B's share will be '21 of 6 times 700 loaves,
makes 200 loaves.
C's share will be '21 of 4 times 700 loaves,
makes 133 '3 loaves.
D's share will be '21 of 3 times 700 loaves,
makes 100 loaves.
Essentially this calculation is carried out in problem
no. 63 of the Rhind Mathematical Papyrus, here given in Eric Peet's
translation: "Example of dividing 700 loaves among 4 men, 2/3 to one, 1/2
to another, 1/3 to another, 1/4 to another. Let me know the share of each of
them." It would hardly be possible to solve such a task properly with 700
loaves placed on several boards in front of you and four men waiting for their
correct share, while it is fairly easily solved in a symbolical way, by using
beans and a further abstraction, bare numbers, which can represent any object
you like.
We are so used to the benefit of numbers that
we can hardly estimate the radically new concept of free numbers, which must
have appeared a strange if not crazy idea to Stone Age man: I can well see
fish; I can even count fish: one fish, two fish, three fish, many fish; but I
can't see any numbers, let alone can I catch numbers, bring them home, have
them cooked and nourish my family ...
Peacock and De Morgan "advocated algebraic
freedom" (Victor J. Katz, A History of Mathematics, Addison-Wesley 1998).
In a similar way one may say that Ahmes in the Rhind Mathematical Papyrus
advocates numerical freedom: Dare consider numbers as beings in their own
right, and you shall be able to solve a variety of demanding problems. Trust
me, and learn from me. Did you solve the problem of the 700 loaves? Did you
like it? Really? So I tell you a secret. I folded many further problems into my
numbers, to be solved by advanced learners. Play with my numbers, and you will
find out what I mean. You need a clue? Well then. Each man got his share of
loaves. Now each one has to pay '8 of his share as tribute, say, to the
fishermen of the village. How do you proceed? You may simply divide the numbers
of loaves by 8. Or you may start from the numerical shares and transform them
into pairs of unit fractions, in such a way that one fraction is the part a man
can keep, and the other one the part he has to pay for the fish, and only then
proceed to loaves. Choosing that way you will find an elegant solution:
'21 of 8 makes '3 '21 '21 of 700 makes 33 '3 loaves
'21 of 6 makes '4 '28 '28 of 700 makes 25 loaves
'21 of 4 makes '6 '42 '42 of 700 makes 16 "3 loaves
'21 of 3 makes '8 '56 '56 of 700 makes 10 '2 loaves
Lesson 40
Let me inform you about a visit to my
astrologer. She kindly allowed me to have a look into her Fine Magic Bablefish
Crystal Ball (trademark reg.), and was able to establish a connection between
me and Ahmes (- with a crystal ball, thousand miles are less than an inch, and
thousand years less than a second, you know). I was pleased to see Ahmes, and
asked him the following question:
Good morning, professor Ahmes, nice to meet
you. May I ask you a question? I run across a funny equation of yours:
1 hekat
times 3 '7 times
'22 of 7 makes 1 hekat
What can this possibly mean? First of all, what
is a hekat? A measure of grain, we know that much, but how is it defined?
Ahmes: Good morning, whoever you are, and
wherever your voice comes from. A hekat is a measure of grain, you are correct.
30 hekat equal 1 cubic cubit, and one hekat is defined as a right
parallelepiped of these numbers:
'2 royal cubit x '3
royal cubit x '5 royal cubit
Me: So simple?
Ahmes: Well, if you understand the basic idea, we
can always go a step further. Consider our subdivision of the royal cubit into
7 palms or 28 fingers or 56 Re marks or 84 Shu marks or 112 Tefnut marks or 140
Geb marks or 168 Nut marks or 196 Osiris marks or 224 Isis marks or 252 Seth
marks or 280 Nephtys marks or 308 Horus marks or 336 Imsety marks or 364 Hapy
marks or 392 Duamutf marks or 420 Qhebsenuf marks or 468 Thoth marks.
Now you can define a square hekat as follows:
28 Re marks
x 28 Shu marks x 28
Geb marks
Or like this:
210 by 140 by 84 Qhebsenuf marks
cubic diagonal 266 Qhebsenuf marks
Me: Amazing. First it looks so simple, and
suddenly we are in the middle of a complex problem. Now tell me what you mean
with your funny number 3 '7 in combination with a hekat.
Ahmes: You are free to play with all my numbers
and inventing problems of your own.
Me: Can it possibly be the number of the
circle? and might there be another hekat
in a round shape, perhaps a cylinder?
Ahmes: Fine guessing. Replace the first hekat
in my equation by the Qhebsenuf definition:
210 Qm x 105 Qm x 84 Qm x 3
'7 x
'7 of 22 = 1 hekat
Now transform the equation as follows:
'4 x 105 Qm x 105 Qm x 3 '7 x
'11 x 3136 Qm = 1 hekat
Can you also guess what the first part means?
Me: Possibly a circle of the diameter 105
Qhebsenuf marks or, wait, 7 fingers? If so, the circle would be the
cross-section of the cylinder, while the second part would be the height. '11 x
3136 fingers are - give me some time - 19 '165 fingers.
Ahmes: Let go the small fraction '165 and keep
the 19 fingers. Thus you got a hekat in the shape of a cup, whose inner
diameter measures 7 fingers, whose inner circumference measures 22 fingers, and
whose inner height measures 19 fingers.
Me: Let me check it --- yes, you are right, the
resulting volume is a hekat, with only a tiny mistake. Thank you for this fine
lesson. By the way ... hello? professor Ahmes? hello?
He was gone. My astrologer failed in getting
another connection. Well then, we shall try again tomorrow.
Lesson 41
My astrologer polished her crystal ball and
connected me again with Ahmes, whom I saw standing by a fisherman in the market
place this time, asking for the price of the fish he just bought, whereupon the
fisherman goes
Fisherman to Ahmes: Give me as much as you
like, but in money, please, not in numbers. You know that I still don't really
understand your profession. Mine is a regular one. I catch fish and sell 'em on
the market. My neighbor is a farmer, he grows emmer, onions, and dates, and
sells them on the same market. We work a lot and just manage to make a living,
while you are in the number business and earn a good living by catching,
planting and selling things nobody can see and touch, let alone cook and eat,
and nourish a family with. I understand well the meaning of one fish, two fish,
three fish, but I still don't grasp the meaning of a number by itself. What can
it possibly be good for? Explain me, professor, and I shall give you a free
extra fish, one of the red ones over here, if you please ...
Ahmes, to the fisherman: Are you making fun of
me?
Fisherman, to Ahmes: Come on, professor, you
know I like you and your family, and I always put my very best fish aside for
you. Only thing is I do not understand what you are doing. I see well that a
man can make a living from fish and grain, but how can a man possibly make a
living from things nobody can see and touch and smell and cook and eat? I
should perhaps ask the gods to send me invisible fish to catch, for they sell
at much higher price than the ones I offer here ...
Ahmes, to the fisherman: You old rascal are
trying to entangle me in yet another discussion on 'number theory' of yours,
until, finally, I see no other way out than paying you some extra money for
your fish. Which is very fine fish, that much is true. My wife always sends me
to buy it from you, because it is the best. You are working hard for making a
living, and I always pay a good price. Here is the money for the fish, and here
the usual extra money for making me laugh. Some day I shall take you with me to
my seminary and let you have a discussion with my pupils. I bet they will have
a hard stand arguing with you on the topic of visible fish, the ones you catch
and sell, and invisible fish, as you kindly call my numbers. I am sure they can
learn from your direct and colorful way of arguing. But now let me go, please.
Fisherman, to Ahmes: Thank you ever so much,
professor, and have a nice day, also in the name of my wife and five children.
Ahmes, to the fisherman: Goodbye, goodbye ...
I listened quietly and hoped to exchange a few
words with the professor upon his returning home, but alas, my astrologer's
crystal ball got blind again. I would have liked to mention how the idea of
pure numbers evolved over time, how 'nothing' became a number, namely zero; how
the negative numbers were introduced, then so-called imaginary numbers, complex
numbers, quaternions and octonions, and even surrealistic numbers; how all the
new numbers puzzled people, and that the concept of number will probably evolve
ever further, causing much more puzzlement in the future, and then, within a
relatively short time, each new number will turn out to be very useful, in one
way or another, whereupon it will be fully accepted within the mathematical
community. Allow me one more word. New expansions of numbers may be expected
where the free flow of natural numbers on any level and the free unfolding of
number patterns, which are the 'footprints' of operations working, are getting
stopped by the restriction of a convention. A thorough study of all kinds of
number patterns may well enhance the further progress in mathematics.
Lesson 42
My astrologer lightly touched her crystal ball,
murmuring some secret words. I got in contact with Ahmes again and asked him
about the meaning of problem no. 23 in the Rhind Mathematical Papyrus.
Me: Good morning, professor Ahmes. May I ask
another question? Sometimes you give us a clue as to what you have in mind, for
example when you mention a hekat. On other occasions you tell us only the
numbers, for example in the case of this problem:
'4 '8 '10 '30 '45 plus
??? makes "3
'4 '8 '10 '30 '45 plus '9 '40
makes "3
Ahmes: Well, you already solved my problem.
Me: Nothing more about it?
Ahmes: I told you before that you are free to
invent any other problems to my numbers. But if you ask me for a clue I would
suggest that you multiply the above numbers by a factor of 360, thus you will
obtain
90+45+36+12+8
plus 40+9 makes
240
191
plus 49 makes
240
Now consider the three numbers as diameters of
three circles and calculate their circumferences, using the value 3 '7 we have
used before, and, very important, rounding all results.
Thus you get the following numbers:
191 x 3 '7
makes 573 + 27 (rounded) makes
600
49 x 3 '7
makes 147 + 7
makes 154
240 x 3 '7
makes 720 + 34 (rounded) makes
754
Proceed like this and you obtain three values
for the secret number living in the circle:
'191 of 600
'600 of 191 makes '4 '30 '40 '100
If you know the circumference of a column or a
circular pond and wish to calculate its diameter, multiply the circumference by
the above unit fraction series.
'49 of 154
makes '7 of 22 or 3 '7
We know that value already.
'240 of 754
makes '120 of 377 or 3
'10 '24 or 3 '8 '60
If you know the diameter of a circle, multiply
it by one of the above series, and you obtain the circumference.
Now I shall inform you about an easy way of
finding many more handy
...
Me: Professor Ahmes? hello? professor Ahmes?
The connection was lost. We shall try again
tomorrow.
Lesson 43
Let me go on with my imaginary dialogue with
Ahmes, scribe of the Rhind Mathematical Papyrus.
Me: Good morning, professor Ahmes. You promised
to show me something ...
Ahmes: Oh yes, I remember. I told you how to
proceed with an equation of mine. Now
have a look at the previous ones:
"3 '15 plus ??? makes 1 "3 '15 plus '5 '15 makes 1
"3 '30 plus ??? makes 1 "3 '15 plus '5 '10 makes 1
Me: I recognize them (Rhind Mathemtical Papyrus
nos. 21/22).
Ahmes: Well then. Multiply the first line by a
factor of 135, and the second one by a factor of 50:
99 plus 36 makes 135 35 plus 15 makes 50
Now multiply the numbers by 3 '7 and round the
results:
99 x 3 '7
makes 297 + 14 (rounded) makes
311
36 x 3 '7
makes 108 + 5 (rounded)
makes 113
135 x 3 '7
makes 405 + 19 (rounded) makes
424
35 x 3 '7
makes 105 + 5
makes 110
15 x 3 '7
makes 45 + 2 (rounded)
makes 47
50 x 3 '7
makes 150 + 7 (rounded)
makes 157
Divide the last numbers by the first ones and give
the results in unit fraction series:
'99 of 311 makes 3 '9 '33 '36 of 113 makes 3 '9 '36
'135 of 424 makes 3 '15 '27 '135 '35 of 110 makes 3 '7
'15 of 47 makes 3 '10 '30 '50 of 157 makes 3 '10 '25
Me: Five new values for the number of the
circle?
Ahmes: Quite so. Do you remember the ones from
yesterday?
Me: '600
of 191 '49 of 154 or '7 of 22 '120 of 377
Ahmes: Now let me come back on my promise and
show you an easy way of how to find all the above and many more handy values
for the secret number living in the circle.
Write 4 above 1 and add repeatedly 3 above 1:
4
(plus 3) 7 10
13 16 19
22 25 28
1
(plus 1) 2 3
4 5 6
7 8 9
Write 3 above 1 and add repeatedly 22 above 7:
3
(plus 22) 25 47
69 91 113
135 157 179
201 223 245
1
(plus 7) 8
15 22 29
36 43 50
57 64 71
78
267
289 311 333
355 377 399
421 443 465
487 509 531
85
92 99 106
113 120 127
134 141 148
155 162 169
Write 6 above 2 and add repeatedly 22 above 7:
6
(plus 22) 28 50
72 ... 424
... 600
2
(plus 7) 9
16 23 ...
135 ... 191
Write 21 above 7 and add repeatedly 22 above 7:
21
(plus 22) 43 65
... 2463
7 (plus 7)
14 21 ...
784 = 28x28
Write 9 above 3 and add repeatedly 19 above 6:
9
(plus 19) 28 47
66 ... 256
3
(plus 6) 9
15 21 ...
81
If you solve a problem that involves a circle
or a sphere you may simply choose a value that comes handy. A few examples:
If a square measures 10 by 10 royal cubits or
70 by 70 palms, the diagonal measures practically 99 palms, and the
circumference of the circumscribed circle measures practically 311 palms.
If the diameter of a circle measures 30 palms
or 120 fingers, the circumference measures practically 377 fingers.
Let the radius of a circle measure 1 royal
cubit or 28 fingers. The area measures practically 2463 square fingers.
Let the radius measure 13 palms. The area
measures practically 531 square palms.
Let a rectangle measure 14 by 16 palms. The
area measures practically 355 square palms.
Square a circle using the value '81 of 256 and
you will find a formula for beginners: a circle of the diameter 9 fingers or
palms or cubits and a square of the side length 8 fingers or palms or cubits
have about the same area.
Me: Thank you, professor Ahmes. Next week I
shall ask you what numbers are ...
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