In my latest book Sacred Number and the Lords of Time, a case is made for numeracy having evolved from (a) late stone age day tallies, recording periods of lunar visibility and invisibility (in which marks were unevenly separated,) to (b) the situation found in the megalithic, in which marks are like those found on a measuring ruler where marks have a defined length between each other, for example an inch, quarters, halves, eighths and sixteens of an inch or centimeters and millimeters. This step turned a habit of making day tallies in the upper palaeolithic, into a powerful system of measures that we call metrology.
Here we focus on how we can grasp numbers held as lengths when modern heads are full of an arithmetic notation which only evolved in the early historical period, in the ancient near east. Today numbers are largely notated, as in 365 days in a year whilst a prehistoric metrology would have to store this number, 365, as 365 units of length making up a single length.
To make this clearer it is worth defining three types or levels of numeracy so as to provide some kind of linkage between the simple numeracy of mere collections and the arithmetic numeracy of notation. The bridge between the stone age and ancient world was a type of numeracy found at the beginning of the ancient near east through which their "megalithic" monuments were achievable and symbolically intelligible only due to an advanced form of metrology. The only intervening culture which could have developed this metrology is the megalithic era, whose role in this has been hidden due to an inability to comprehend the significance of this second level of numeracy and of what it was capable.
To approach numeracy without arithmetic involves maintaining numbers within the physical world as a type of media. One sees such media in stone age artifacts as decoration such as tally counts and geometrical patterning or as similar objects such as beads strung to form a necklace. One of the best media for exploring stone age numeracy is blank dice which are available on the web, in different colours. My initial experiments quickly gave a completely different view of familiar situations within geometry, such as the 3-4-5 "Pythagorean triangle. Below shows the dice workspace in blue.
Dice Workspace for experimenting with pre-arithmetic numeracy.
The use of a cube is ideal for experimenting with number held within lengths since line, area and volume of geometrical dimensionality can be revealed with conservation of those properties. For example, when one draws a rectangle, whose area is an implicit multiplication of length times width, a line is drawn as a border of the enclosed space. Using dice, the area can be filled with unitary objects so as to show the internal area, not in outline and to be calculated, but as the resulting square area. Two times six results in twelve objects. The area of one squared is one, of two squared four, of three squared is nine, four squared sixteen and five squared twenty five, forming a stepped pyramid of volume 1 + 4 + 9 + 16 +25 = 55, as shown above.
One can see the late stone age at work with the properties of line area and volume in Megalithic Brittany (see later) and copper age Portugal: "Cachão da Rapa (Trás-Os-Montes, Portugal) was the first rock art site described in the Iberian Peninsula (Carvalho da Costa 1706), and one of the pioneers in Europe. It is considered to be among the most original paintings of the Iberian schematic rock art" from Cachão da Rapa prehistoric rock art paintings revisited. My own analysis of its drawings is as below and appears to show a proof of Pythagoras for the 3-4-5 triangle when approached using non arithmetic numeracy.
Analysis of Etching by Debric of paintings in the rock shelter of Cachao da Rapa
One can see a predominance of squares based on different numbers, using a unitary square size. I have colour coded the areas shown in the etching and brought out the geometrical groups so as to propose a sequencial thinking with regard to the more complex forms made of squares which hold twenty and twenty five squares. Sometimes a column or row is shown as a whole rather than a set of squares. The squares of two, three, four and five are shown plus areas 3 x 4 = 12 and 2 x 3 = 6. I propose that the 3-4-5 triangle was seen, via the rectangle 4 x 3, to have a diagonal length five through a rational in which, four squared and three squared "fit" within five squared to have (a) six squares overlapping which can then be distributed to (b) the six squares unfilled by those two squares. This means that 3 x 3 plus 4 x 4 equals 5 x 5 whilst the side length of 5 equals the diagonal of the 4 x 3 rectangle. The fingers and thumb of a hand appear related to the unitary squares as if also used within this pre-arithmetic "thinging".
The 3-4-5 triangle would prove very significant to the Megalithic, most especially the astronomers at Carnac in Brittany, France during the 5th and 4th millennium BCE where the sun at that latitude actually generated the acute angle of a 3-4-5 triangle relative to east-west, at summer and winter solstice, sunrise and sunset.
The Extremes of the Sun relative to East
was aligned to the five side of the 3-4-5 triangle,
at the latitude of Carnac in the 5th millennium BCE
Between the solstice and equinoctal sunrises there are just more than three lunar months, the number of units on the three side of the triangle above. In a complete year, solstice to solstice, there are twelve lunar months [3 + 3 + 3 + 3], that is the number of whole lunar months within a solar year, plus just over one third of a lunar month [10.875 days]. (the lunar month being 29.53 days long on average). These twelve whole months are easily symbolised as being the twelve whole "cubes" in the 4 x 3 rectangle.
This means that in three years there are 3 x 12 = 36 lunar months plus just over one extra lunar month [37.1]. The thirty six whole months are then "like" the cubes in three 4x3 rectangles:
And these can be brought together to form a 12 x 3 rectangle of 36 lunar months:
And what then happens is that the diagonal length of the 12 x 3 rectangle relative to the length of its twelve side is the length of the solar year in lunar months, which equals 12 and 7/19 [12.368] lunar months. It is also clear that a 12 x 3 rectangle is a four square rectangle with side length equal to three:
One sees a play between the factors of twelve, 4 and 3, in both this assembage of three 4 x 3 rectangle, becoming four 3 x 3 squares, and three and four being the side lengths of the 3-4-5 rectangle at Carnac, describing the range of the sun.
Relation of three 4 x 3 rectangles as 36 lunar months,
and a 12 x 3 rectangle as a four-square rectangle
whose diagonal is then three solar years long, in lunar months.
In the above diagram, the mind has not become separated from the structure of the problem as belonging to a physical, pre-arithmetical process of transformation. By taking three 4 x 3 rectangles, joining them into a 12 x 3 rectangle of 36 square units, the period of three solar years marks a near anniversary of the solar year with the lunar month and the geometry can be reconfigured as being four squares of side length three months, which can be tilted towards the solstice. The tilt required is the 36.8 degrees north of east minus 14 degrees of the four-square diagonal, giving 22.6 degrees of the "second Pythagorean triangle" or diagonal angle of a 12 by 5 rectangle:
The amount of tilt required to make the diagonal
align to the solstice sun is the diagonal angle of a 12 x 5 unit rectangle
whose diagonal is then exactly thirteen units long.
It is now possible to see the Le Manio Quadrilateral as an expession of this sort of thinging in which the southern kerb of 36-37 stones east of the sun's "solstice gate" are angled according to the diagonal of a 12 x 5 rectangle and in which the three solar year count was recorded between that gate's mid-point and the sculpted edge on stone R, counted in day-inches as 1095.75 day-inches long:
The orientation, size and shape of the Quadrilateral at Le Manio follows the sort of process followed above, from first principles of simple "unitary" facts regarding numbers and geometry. It may be that megalithic activities would always be couched within the form of such a thinging process, exactly because they could not abstract with notation the numerical facts from the thing being measured, hence their use of metrology as numbers expressed within lengths rather than as pure symbols, as were developed later in the ancient near east.
The destruction of megalithic monuments over millennia and our own changed mechanisms of thinking about problems requires both the finding of perfect surviving monuments recording such prehistoric thought forms and an equal effort to escape our own habitual mechanisms, by learning how to work with numbers as they did, as objectified collections of units.
More about Le Manio is in print as my book Sacred Number and the Lords of Time, whilst my brother and I have published a pdf report on academia.edu as The Origins of Megalithic Astronomy as found at Le Manio.