Natural time periods between celestial phenomena hold powerful insights into the numerical structure of time, insights which enabled the megalith builders to access an explanation of the world unlike our own. When looking at two similarly-long time-periods, the megalithic focussed on the difference between them, this causing the two periods to slide in and out of phase, generating a longer period in which the two celestial bodies exhibit a complete ensemble of variation, in their relationship to each other. This slippage of phase between celestial periods holds a pattern purely based upon number, hidden from the casual observer who does not study them in this way. Such numerical patterns are only fully revealed through counting time and analysing the difference between periods numerically.

For example, the solar year is longer than the lunar year by 10 and 7/8 days (10.875 days) and three solar years are longer than three lunar years by three times 10.875 days, that is by 32 and 5/8th days (32.625 days), which is 32/29 of a single lunar month of 29.53 days.

*Figure 1 (in plan above) The monumentalising of a three-year day inch count at Le Manio as a right triangle based upon its southern kerb (in profile below), automatically generating the megalithic yard.*

The earliest and only explicit evidence for such a three-year count has been found at Le Manio’s Quadrilateral near Carnac (circa 4,000 BCE in Brittany, France), which employed the inches we still use to count days, a “day-inch” unit then widespread throughout later megalithic monuments and still our inch, 1/12 of the foot [Heath & Heath. 2011]. The solar-lunar difference found there over three years was 32.625 day-inches, this length the probable origin of the unit we call the megalithic yard, coined by Alexander Thom. The megalith builders seem to have adopted this differential length, between a day-inch count over three lunar and solar years, as a measure for building many later monuments.

Le Manio’s Quadrilateral (figure 1) shows that the geometry of the right triangle was employed to express and manipulate lengths of time as day-inch counts of lunar and solar years. In such a triangle, the difference is revealed between its two longer sides, a difference most clearly seen between the right angle of the triangle (at stone 36 on its base) and the end of the longest side “arced down” over the base (stone 37) using ropes[1]. This longest side, symbolically aligned to the summer solstice sunrise, contains no stones: implying that the solar count was used to inform the subsequent construction of the triangle between lunar and solar lengths, only then to build the southern kerb of stones whose exact positions and shapes, we shall see, then corresponding to other time periods within three years that were deemed significant (see figure 2).

*Figure 2 The southern kerb of the Quadrilateral in silhouette, looking south, ** based on a calibrated photo montage. Note that western edge of stone 35 is located where three eclipse years would end when counted from stone 1.*

If one makes the differential length into a megalithic yardstick, one can count the number of those yards in the base and longest sides of the triangle as being 32.625 and 33.625, that is as being just one megalithic yard different. This is a powerful feature of right triangles, that their differential length always divides into its longest sides to “normalise” the triangle, as being an N:N+1 triangle. It is also true that whatever the number of years over which lunar and solar years are counted, the triangle will be self-similar (of the same form), having side ratios of 32.625:33.625. By good fortune, the megalithic noticed the short near-anniversary over three years of 36 and 37 lunar months, so giving them the invariant N = 32.625, as the difference – which it need not have been! This led me to the conclusion, in my books, that such good fortune facilitated the arising of megalithic astronomy at a time when Neolithic activities were thought only to have been evolving early forms of agriculture, rather than an advanced astronomy based upon counting time.

Le Manio therefore represents the foundational application of time counting, of the two main luminaries, by studying numeric differences to systematise time. Within the three year frame of synodic time lie other interesting phenomena including the lowest number of lunar months that occurs on an integer number of solar days: In 32 lunar months there are exactly 945 days, to within half an hour. This allowed an accurate approximation to the actual length of the lunar month if 945 days are divided into 32 parts, each one therefore being 29.53125 day-inches long. This compares to the true average lunar month of 29.53059 days, to one part in 44743 and so is therefore, effectively exact. But to access such an estimate as a length, there had to be a megalithic technology capable of dividing 945 by 32.

There is evidence in the late stone age for the study of how numbers interacted, using objects like beads and etched patterns on bones and rock surfaces. It is quite possible to factorise a long count by using a measuring rod to count: if there is no partial excess then the length of the measuring rod (in inches) is a factor of the number in day-inches. The factors of 945 day-inches are 3^{3} x 5 x 7. A cube 27 by 5 by 7 inches would contain 945 cubic inches. A technique was then developed which I call “proximation”, where two similar numbers with suitable factors can be placed within a right triangle – enabling the geometrical equivalent of division, given suitable factors within the two numbers.

Those factorising in this way would know 32 as the product of five twos (2^{5}) and would also see *within* 945 the factors 9 x 7 as those making up 63, just one less than the twice the 32 lunar months, and the 64 is the product of six twos (2^{6} = 64). Divided into 63 parts, each part of 945 day-inches would be 15 day-inches long, whilst 64 x 15 = 960, the longest side over the base of 945 in a right triangle.

*Figure 3 Exploiting the factors of 945 day-inches to rescale and resolve an accurate length for the lunar month*

However, by using 64 one has doubled 32, and we are looking for a proximate length to 30 day-inches rather than the natural factor 15 day-inches. Therefore, they looked at the “sharp” end of the triangle (see figure 3) and saw above (on the hypotenuse) two 15-day periods equalling 30 days. The base of the triangle (shown in yellow) is then automatically 64/63 x 30 day-inches, which equals 29.53125 day-inches, the lunar month. That is a 63:64 triangle was possible with hypotenuse equal to 30 day-inches and base 29.53125 day-inches, but this size was a bit too small for convenience.

The fractional part of 29.53125 day-inches is 17/32 day-inches but measuring to 32^{nds} of an inch was not practical so instead; they *quadrupled* the counting units by generating the length of four lunar months, i.e. a hypotenuse length of 120 day-inches or ten feet (now called a double-fathom or perch). This rendered the size of the fractional part of the base length, using a four-times bigger triangle, 17/8th or 2 plus 1/8th. This length, of 118.125 inches long is the number of day-inches in *four* lunar months to high accuracy and this would have been a familiar counting length for megalithic astronomer in this earliest phase, since two months are about 59 day-inches long, cancelling the half day in 29.53 day-inches. Doubling again to 118 1/8th day-inches (118.125) resolves the more practical factional measure of 1/8th day-inch (0.125).

We can see the 32 lunar month period in the end of the 31st stone of the southern kerb at Le Manio. From the left-hand edge of stone 31, to the gap between the solstice “Sun Gate”, measures 945 inches. Then there is a gap filled by stone 32 which is narrow. Why make stone 32 narrow with gaps? Firstly, one should point out it could have been laid later, in imitation of stone 35.

*Figure 4 Calibrated photo survey conducted of Le Manio’s Southern Kerb in 2010.*

I believe stone 32 sits in a gap representing the 0.625 of a lunation between 32 and 32.625 lunar months. The reader may remember that 32.625 is N for the normalised N: N+1 relationship of the triangle formed between three lunar years and three solar year. Over three lunar and solar years, this manifest as a single whole lunar month (in angle) of the sun on the ecliptic, when the sun is illuminating the 32nd lunar month. That is, the sun is moving whilst the moon is being illuminated by it. The 33rd stone of the southern kerb therefore represents the whole lunar month between 32.625 and 33.625 months in the day-inch count, whilst the 32nd stone, with unusual gaps either side, is narrow (from starting at 32 months in the count) to represent 0.625 of a lunar month.

That is, the 33rd stone was intended to be the lunar month[2].

Also in the frame of Figure 4 is stone 29 and above we stated:

the solar year is longer than the lunar year by 10 and 7/8 days (10.875 days) and three solar years are longer than three lunar years by three times this, 32 and 5/8th days (32.625 days), which is 32/29 of a single lunar month of 29.53 days.

The gap between stones 32 and 29 relative to the “sun gate” represents the normalising difference which resolves, somewhat miraculously, to the numbers 32 and 29 in relating the astronomical megalithic yard[3] (32 units of 1.0183 inches) to the lunar month (29 of 1.0183 inches); in a day-inch count over three years.

*In part two I will look at the overall form of the monument as an integrated expression of synodic spacetime.*

## Summary and Conclusions

In counting order therefore, the end of the southern kerb (if we read it numerically) at least portrays within its choice and disposition of stones,

- the difference between the lunar month and the three-year differential equal to the megalithic yard in day-inch counting.
- the 32 lunar month equivalence to 945 days which gives a highly accurate estimate of the lunar month of 118 and 1/8th day-inches divided by four (29.53125).
- the gap of 0.618 months between 32 and 32.625, N for the solar-lunar ratio over three years.
- Three eclipse years from the start to the westernmost edge of stone 35 (1040 day-inches).
- Three lunar years from the start to stone 36 (1063 day-inches), numerically 3 x 12 lunar months.
- Three solar years from the start to the end of stone 37 (1096 day-inches).

The stones of the Quadrilateral appear to have been laid to form a numeric sketch pad of key relationships visible within its range of three solar years of day-inch counting.