46 Lessons in
Early Geometry, part 1/10 / provisional version in freestyle English / a
corrected version will follow in March, April or May (hopefully) / Franz
Gnaedinger / February 2003 / www.seshat.ch

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46 Lessons in Early Geometry

Lesson 1

The caves of the 'Ile de France' (

Imagine a shaman drawing a grid of, say, 4 by 4
squares on the fine clay surface of a river bank. The grid will contain the
squares 1x1, 2x2, 3x3, 4x4. Measuring their diagonals the shamen may find the
following relations:

If
the side measures 1 unit, the diagonal measures about one and a half unit

If
the side measures 2 units, the diagonal measures about 3 units,

and
if the side measures 3 units, the diagonal measures about 4 units

The shaman will test his numbers by drawing
larger grids, and he will modify the results as follows:

If
the side of a square measures 2 units, the diagonal measures a little less than
3 units,

and
if the side measures 3 units, the diagonal measures a little more than 4 units

What if he drew a square of side length 2 + 3 =
5 units? The diagonal of this larger square should measures a little less than
3 units plus a little more than 4 units. The "little less" and
"little more" may even out, and the result will be 7 units.

side
5, diagonal 7 side 7, diagonal 10

By drawing a still larger and more precise grid
he will realize that one diagonal is slightly shorter, and the other one
slightly longer, so he will proceed in the same way again:

side
12, diagonal 17 grid side 17, diagonal 24

This time the result seems to be fine enough
for practical purposes, but what if one diagonal is again slightly longer and
the other one slightly shorter? and what if the same is true of all subsequent
diagonals?

If so, one could approximate the diagonal of
the square by means of the following number pattern. Double the first number of
each line and you obtain the last number, add a pair of numbers and you obtain
the number below:

1
1 2

2
3 4

5 7
10

12 17
24

29
41 58

70 99
140

If the side of a square measures 29 paces, the
diagonal measures 41 paces, and if the side of a square measures 70 paces, the
diagonal measures 99 paces.

Lesson 2

The side of a square field measures 63 paces.
If you are asked to calculate the diagonal, you may proceed as follows:

side 63 = 41 + 17 + 5

58 + 24 + 7 = 89 diagonal

side 70 - 7

99 - 10 = 89 diagonal

The exact number would be 89.095...

Let the side of another square field measure 79
paces. How long is the diagonal?

79 =
70 + 7 + 2

99 + 10 + 3 = 112 diagonal

side 79 = 6x12 + 7

6x17 + 10 = 112 side

The diagonal measures 122 paces (exact number
111.722...)

Let a square measure 400 by 400 cubits. How
long are the diagonals?

side 400 = 2x70 + 2x99 + 41 + 3x7

2x99 + 2x140 + 58 + 3x10 = 566

side 400 = 3x70 + 99 + 41 + 41 + 7 + 2

3x99 + 140 + 58 + 58 + 10 + 3 =
566

The diagonal measures 566 cubits (exact number
565.685...)

Let the diagonal of another square measure 729
meters. How long are the sides?

diagonal 729 = 3x140 + 3x99 + 3x4

3x99 + 3x70 + 3x3 = 516

diagonal 729 = 2x140 + 4x99 + 41 + 12

2x99 + 4x70 + 29 + 17/2 = 515 1/2

The side measures 516 or 515 1/2 cm (exact
number 515.480...)

Lesson 3

A number pattern can be started with any pair
of numbers, and you are allowed to make an occasional mistake*:

1
3 2

4
5 8

9
13 18

22 31
44

53 75
106

128 181
256

If the side of a square measures 128 cubits,
the diagonal measures 181 cubits (exact number 181.019...)

2
7 4

9
11 18

20
29 40

49 69
98

118 167
236

If the diagonal of a square measures 236
meters, the side measures 167 meters (exact number 166.877...)

1
1 2

2
3 4

5 6*
10

11 16
22

27 38
54

65 92
130

157 222
...

If the side of a square measures 157 cm, the
diagonal measures 222 cm (exact number 222.031...)

Lesson 4

Let the edge of a cube measure 41 fingers. How
long are the diagonals of the faces? and how long are the cubic diagonals?

1
1 2

2
3 4

5 7
10

12 17
24

29 41
58

If the edge of a cube measures 41 fingers, the
diagonal measures 58 fingers (exact number 57.982...)

Now let me draw up an analogous number pattern,
using a factor of 3, and dividing all three numbers of a line by 2 whenever
possible:

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

If the edge of a cube measures 41 fingers, the
cubic diagonal measures practically 71
fingers (exact number 71.014...)

The second number pattern allows the
calculation of the cube, of the equilateral triangle, and of the regular
hexagon.

Lesson 5

Draw a small square or a small rectangle 'a.'
Add square 'b' to a longer side of 'a,' thus you obtain rectangle ab. Add
square 'c' to a longer side of ab, thus you obtain rectangle abc. Add square
'd' to a longer side of abc, thus you obtain rectangle abcd. And so on golden rectangle
Beginning with square a = 1x1 and moving in clockwise direction you may
obtain this rectangle:

f f f f f f f f g

f f f f f f f f

f f f f f f f f

f f f f f f f f

f f f f f f f f

e e e e e b c c

e e e e e a c c

e e e e e d d d

e e e e e d d d

e e e e d d d d

The resulting figures will approximate the
golden rectangle (ratio Phi =
1.6180339...). The numerical equivalent of the above drawing procedure are
golden sequences that can again be started with any pair of numbers and again
allow an occasional mistake*:

1 + 1 = 2

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

1 + 3 = 4 13 + 21 = 34

3 + 4 = 7 21 + 34 = 55

4 + 7 = 11 34 + 55 = 89

7 + 11 = 18 55 + 89 = 144

11 + 18 = 29

2 + 7 = 9 18 + 29 = 47

7 + 9 = 16 29 + 47 = 76

9 + 16 = 25 47 + 76 = 123

16 + 25 = 41 76 + 123 = 199

1 + 1 = 2 25 + 41 = 76

1 + 2 = 3 41 + 76 = 123

2 + 3 = 5 76 + 123 = 199

3 + 5 = 9* 123 + 199 = 322

5 + 9 = 14

9 + 14 = 23

14 + 23 = 37

23 + 37 = 60

37 + 60 =
97

60 +
97 = 157

Lesson 6

Let the side of a square and the edge of a cube
measure 50 fingers each. Double the area of the square and the volume of the
cube.

The side length of the larger square is given
by the diagonal of the initial square:

side 50 = 41 + 7 + 2

58 + 10 + 3 = 71 diagonal

side 50 = 4x12 + 2

4x17 + 3 = 71 diagonal

If the side of a square measures 50 fingers,
the side of a square of the double area measures about 71 fingers (exact number
70.710...)

1
1 1
2

2
2 3 4

4 5
7 8

9 12
15 18

3 4
5 6

3
4 5 6

7
9 11 14

16
20 25 32

36 45
57 72

12 15
19 24

12
15 19 24

27
34 43 54

61
77 97 122

138 174
219 276

46 58
73 92

46
58 73 92

104
131 165 208

235
296 373 470

531 669
843 1062

177 223
281 354

177 223
281 354

400
504 635 800

50
63

If the edge of a cube measures 50 fingers, the
edge of a cube of the double volume measures practically 63 fingers (exact
number 62.996...)

Lesson 7

Do you remember the shaman of my first message?
He drew a grid of 4 by 4 squares on the fine clay surface of a river bank and
measured the diagonals of the squares 1x1, 2x2, 3x3 and 4x4.

The grid 4x4 also provides the rectangles 1x2,
1x3, 1x4, 2x3, 2x4 and 3x4. By measuring their diagonals the shaman will find a
simple result: if a rectangle measures 3 by 4 units, the diagonals measure 5
units.

By drawing larger and more precise rectangles
he will confirm his result: the diagonals really seem to measure 5 units, or
the mistake is so very small that I can neglect it for practical purposes.

If our shaman lived some 6500 years ago in

A . . . . . . . B

.
N . AC = BD = 3 units

. .

.W E. AB = CD = 4 units

. .

.
S . AD = CB = 5 units

C . . . . . . . D

CA and DB
running south-north

AB and CD
running west-east

Sighting lines:

AB or CD - rising sun on March 21

BA or DC - setting sun on March 21

CD or AB - rising sun on September 23

DC or BA - setting sun on September 23

CB - rising sun on June 21

DA - setting sun on June 21

AD - rising sun on December 21

BC - setting sun on December 21

A stone rectangle of these proportions is found
at Plouharnel,

Lesson 8

Among Swiss archaeologists the northern shore
of

In my opinion, the first sanctuary consisted of
seven menhirs that have been placed there in around 4300 BC or 6300 BP and
combined a large raven (corvus corax) with a solar calendar and a midsummer
corridor:

B

A

G

C

E

F

D

Seen as a large raven, Menhir E marks the head,
menhir C marks the body, menhir A marks the tail, and the menhirs FD and GB
mark the open wings.

The five menhirs A B C D E would have served as
a calendar in the shape of an oblique sandglass (that also represents a cycle
of vegetal, animal and human fertility):

A
winter (December 21)
B

B
spring (March 21) A

C
Beltane (May 1) C

D
summer (June 21) E

E fall
(September 23) D

C
Samhain (November 1)

ACE was the sighting line of the midwinter sun
rising above the near

The angle of the lines DG and FE in relation to
the pole star (then in Draco) had a tangent of practically 4/3, as in

The seven menhirs A-G seem to occupy an
imaginary grid that measures 14 by 12 large units (about 516 cm each) or 84 by
98 small units (about 73.7 cm each; nine small units may equal eight of
Alexander Thom's "Megalithic Yards").

Menhir C (Beltane/Samhain) marks the center of
the grid; menhirs B (spring) and D (summer) mark its northeast and southwest
corners. The four calendar menhirs A B D E were placed symmetrically to the
central menhir C, while the parallel lines DG and FE of the midsummer corridor
have a tangent of 3/4 in relation to the grid's west-east axis. The angle BDE
doubles the angle BDG and equals angle ABD.

The geometrical drawing requires only ropes and
poles, a grid, diagonals, and two circles for doubling the angle BDG, while
yielding very interesting numbers (tangent of the line DCB 7/6, tangent of the
lines DG and FE 3/4, tangent of the angle ABG 2/9; the prolonged line BA
measures 105 small units according to the triple 3-4-5 or 63-84-105 and divides
the western side of the grid 98x84 into 60+38=98 small units, while the
prolonged lines DE and DG divide the grid's eastern side into 38+25+35=98 small
units; the long diagonal BCD measures practically 129 small units, the line ECA
measures practically 57 small units, and the parallel lines AB and DE measure
practically 59 Megalithic Yards each).

Menhir chapters in German, with 200
illustrations menhir a / menhir b / menhir c

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