Table des Marchands, a dolmen at Lochmariaquer, can explain how the Megalithic came to factorise 945 days as 32 lunar months, by looking at the properties of the numbers three, four and five. At that latitude, the solstice angle of the sun on the horizon shone along the 5-side of a 3-4-5 triangle to east and west, seen clearly at the Crucuno Rectangle. But in the dolmen's carving these numbers could have been put to another good use.

Before numbers were individually notated (as with our 3, 4 and 5 rather than |||, |||| and |||||) and given positional notation (like our decimal seen in 945 and 27), numbers were lengths or marks and, when marks are compared to accurately measured lengths measured out in inches, feet, yards, etc. then each vertical mark would naturally have represented a single unit of length. This has not been appreciated as having been behind marks like the cuneiform for ONE; that it probably meant "one unit of length".

*Figure 1 The end and cap stone inside the dolmen Table des Marchands in which the elementary numbers in columns and rows perhaps inspired its local attribution to the accounts of merchants*

In the carvings of the end stone (C4) of Table des Marchands, groups of crook shaped lines were created in which the crooks point away from the central axis of a stone whose top section is an oblate isosclese ("equal legged") triangle whose base angles are 60 degrees, within which the array of grouped crooks appear. The outer oblate edges are carved in a grooved border which looks like a long count. Below the triangular section is a rectangular panel of detailed drawings and/or symbols whose meaning is now probably lost forever.

## Early Number Systems

The mark for 1 in cuneiform (meaning "wedge shaped", from 3000 BC in Near East) had a triangle atop, since the reed used in small writing naturally made that and indeed, it makes cuneiform visually striking. However, every "1" is then the same and so why were the crooks arranged with crooks opposed, to either side? And why were they in number pairs which appear to follow a regime of N to the left then N+1 to the right?

The row below N will have N+1 (repeated) to the left and N+2 to the right and that regime followed as we might in fractions where one number is placed on top as the numerator and another is placed below as the denominator (as in 27/25). We know that historical metrology defined its measures as rational fractions of the foot so that, for example, in 8 feet there are 7 Royal feet of 8/7 feet. This habit, whatever other reasons there were for it, allowed a single reference unit to translate a single standard foot into the Royal foot of 8/7 using a right angled triangle where the longest sides were 7 feet (the base) and 8 feet (the hypotenuse). Or, as seems likely, one could just know that in 8 feet there are 7 royal feet, an aggregate unit of length as with a yard or a fathom.

*Figure 2 Triangular and Aggregate ways of Generating known quantities of other feet in ratio to the normal foot.*

Right angled triangles are found throughout the monuments of Carnac, for example explicitly as the triangle of the lunar year to the solar year at Le Manio Quadrilateral or implicitly as the diagonals of multiple squares seen in the plans of dolmens, rectangles and alignments. Indeed, it seems the astronomers had been attracted to the latitude of Carnac (LoT) where, on the horizon, the solstice sun shone along the hypotenuse of a 3-4-5 triangle, the maximum moon standstill along the diagonal of a single cardinal square and the minimum along the diagonal of a double square (noted by AAK and more recently, Howard Crowhurst). Using ratio measures to calculate is strongly associated with the triangles and rectangles within monuments, justifying therefore the interpretation of this crook notation as involving, in some way, ratios and their small number factors of significant numbers.

*Figure 3 The AAK drawing of stone C4 found in volume 20 of their magazine Etude et Travail.*

The rows are numbered

- 4 to 5
- 5 to 6
- 6 to 7 (the six is shown by drawings as seven but two crooks may have once been one, see Twohig's version in figure 4)
- 7 to 8

The top appears eroded or shaved off. I believe this place had the number 3, looking a bit like the fleur-de-lis. It was placed on the central axis in which the central crook was straight whilst the side crooks were opposed, so setting the pattern in which the crooks either side were opposed. The top of the end stone would have formed a triangle of the numbers 3, 4 and 5:

*Figure 4 How placing three as a fleur-de-lis form gave access to significant larger numbers using easy multiplications and additions from a root of 3, 4 and 5.*

We now go on a necessary detour, to establish how factorisation could be involved in stone C4 for only then one can understand how the numbers 3-4-5 can operate together, using rudimentary arithmetic, to present a useful mnemonic device, helpful to the astronomers of an oral culture. The data-compression made it possible to present on a stone design instead of through measurement and factorisation. Its subject was how to factorise 945 days so as to divide it by the 32 lunar months in that period.

## Factorization and Metrology

The megalithic had no means of multiplication except through adding a number successively a certain number of times, to make a longer length. Thus, if one had a yardstick one could use it 100 times to obtain 300 feet. And to divide a number into a length of 331 feet by three, one could use the yardstick 110 times and find a single foot left over. Such a method of division effectively removes a single factor out of the length. This seems rather longwinded but better than nothing. And whilst testing for a factor, one would create a count of that counting. This allows for the saving of results not offered by counting using fingers and parts of the body.

Once the 945 days in 32 lunar months had identified as a coincidence/anniversary between whole numbers of days and lunar months, 945 needed to be factorised. Today we look and see 5 at the end of the number and know 5 must be a factor in it. At Carnac there is only a day count length, on the ground. How else to store this and manipulate it except by knowing its factors? With a rod five feet long, there would be 189 rods in 945 feet with nothing left over. To find a further factor of 189, it is helpful to have made the length of 189 feet whilst doing the five foot factorisation, allowing that to be further factorized. When five won't work, a yardstick of 3 feet could be tried, leading to a new count 63 feet long.

*Figure 5 Finding the divisors of 945*

Most numbers under 100 would be known as to their factors so that 63 would be recognised as 3 x 3 x 7 (nine times seven) and the full factorization of 945 clearly 3 x 3 x 3 x 5 x 7 foot measures to make then 945 feet. We would call its factors 3^{3}.5.7 but the more practical needs of the megalithic were to find numbers proximate to each other. The cube of three is 27 and 5 x 7 is 35 and 945 seen as 27 x 35 feet.

It is known 945 days are 32 lunar months and 32 is proximate to 35 and the desired length is 945/32 (the lunar month) as a day length. The use of rational fractions in early metrology is that it allows these sorts of transformation of units of measure according to a measurement's factorization.

Feet were large enough to support the use of a range of fractions, a new type of foot created, in this case, to effect the necessary fraction of 35/32. The effect of dividing ordinary feet by a another rational type such as 32/25 feet is to multiply by the reciprocal, namely 35/32. This is experimentally obvious: dividing a length by the smaller 32/25 makes the original measurement, in feet, a larger number of the lesser feet, of 32/35. The only other factor is 27 so that when one divides 27 feet by 32/35 feet, the formula of the result is 945/32, the length of the lunar month (29.53125) in the new feet. This is initially hard to understand, but 945 days divided by 32 is 29.53125 days. It is accurate to 57 seconds in more than 2.5 million seconds!

There was then no need to actually count the 29.53125 days implicit within 27 feet but two facts sufficed.

- A day could be measured as a foot of 32/35 feet within a larger count using 27 feet per month and,
- A count of lunar months could be counted using 27 ordinary feet.

This is exactly what I found (using Google Earth) within the Crucuno rectangle and then from Crucuno Dolmen, that 27 foot units were being used to count from the dolmen to within the rectangle as 47 months which functions as the shortest coherent eclipse period, of four eclipse years (see this article).

## Interpreting stone C4 at Table des Marchands

At last one can see why I suggest putting three atop stone C4 so as to form the "primordial" triad of numbers 3-4-5, those corresponding to the 3-4-5 triangle of such utility at Carnac, but now treated as a means to remember the solution of using 27 feet to represent the lunar month of 29.53125 days. The factors of the 945 day period are 27 and 35 whilst 32 is the number of lunar months in that period.

- The numbers can be multiplied in three pairs to yield 12, 15 and 20 (see right hand diagram in figure 6).
- These numbers can be added, in three pairs, to yield the needed factors of 27, 35 and 32, the factors built into the relationship of 945 days in 32 lunar months.

When 27 is multiplied by 35/32 it must get bigger and it becomes 29.53125, the lunar month length implied by the 945 co-incidence with 32 lunar months.

*Figure 6 Figure (right) How the numbers derived from 3, 4 and 5 yield the factors of the 945 day period of 32 lunar months. (left) The remaining numbers supply the ratios of 32/35 and 35/32 as a multiplication table and the ruling axis of 27.*