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

46 Lessons in Early Geometry

Lesson 1

The caves of the 'Ile de France' (Paris and the region around it) abound of geometric rock incisions from Paleolithic, Mesolithic and Neolithic times, among them many squares and square grids

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 Brittany he could have used such a rectangle on top of a flat hill for observing  the sun and measuring time:

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, Carnac. Another one at Finisterre had been removed, yet an old drawing shows that one of the short sides was marked by seven stones yielding 6 wide spaces, while one of the long sides was marked by nine stones yielding 8 wide spaces.

Lesson 8

Among Swiss archaeologists the northern shore of lake Neuchatel is known as "little Brittany," because there are found so many menhirs. A Neolithic sanctuary comprising some 40 menhirs was built on the former shore of the shallow southwest bay of the fairly straight and some 30 km long lake, on nearly the same geographical latitude as Carnac. The first menhirs are dated as early as 4500 BC or 6500 BP. Nowadays the site belongs to Yverdon-Clendy, and the shallow bay was filled up. A mass of reed blocks the view on the lake. Yet a Neolithic visitor had a free view on the water, and gathering there on an early midsummer morning the pilgrims could observe a miracle: the sun rose from the middle of the lake ...

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 Bellevue hill. DG was the sighting line of the midsummer sun rising from the middle of the lake, while the parallel lines DG and FE mark the midsummer corridor.

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 Carnac, allowing the application of the triple 3-4-5. And, if my geometrical examination is correct, this basic triple had been used.

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