In previous posts, it has been shown how a linear count of time can form a square and circle of equal perimeter to a count. In this way three views of a time count, relative to a solar year count, showed the differences between counts that are (long-term average) differential angular motion between sun and the moon’s cycle of illumination. Set within a year circle, this was probably first achieved with reference to the difference between the lunar year of 12 months (29.53 days) and the solar year of 12 average solar months (30.43 days). Note that in prehistory, counts were over long periods so that their astronomy reflected averages rather than moment-to-moment motions known through modern calculations.

The solar year was a standard baseline for time counting (the ecliptic naturally viewed as 365.25 days-in-angle, due to solar daily motion, later standardized as our convenient 360 degrees). Solar and other years became reflected in the perimeters of many ancient square and circular buildings, and long periods were called super years, even the Great Year of Plato, of the precession of the equinoxes, traditionally 25920 years long! The Draconic year, in which the Moon’s nodes travel the ecliptic, backwards, is another case.

Above is a proposed geometric relation between Thom’s megalithic yard (2.72 feet), the royal cubit (1.72 feet) and the remen (1.2 feet). Alexander Thom’s estimate for it based on decades of work was refined from 2.72 to 2.722 feet at Avebury. If the origins of it are astronomical, then its value emerges from the Metonic period of 19 years which is 235 lunar months, making its value 19/7 feet or more accurately 2.715428571 (19008/7000) feet and this makes it 2.7 feet x 176/175 within ancient metrology. Another astronomical derivation is found at Le Manio as the difference between three lunar and three solar years, when counted in day-inches as 32 + 5/8th inches which is 2.71875 (87/32) feet. The megalithic yard of Thom’s first appraisal, of 2.72, probably arose from its megalithic rod (MR) of 6.8 feet since, the Nodal Period of the moon’s nodes take 6800 days which in feet would be 1000 MR. For a fuller explanation see my the appendix of my Language of the Angels book and my discussions of the Cumbrian stone circle, called Seascale by Thom and the only known example of a Type D flattened circle.

One can see that the Megalithic Yard is a tale of many variations, some of which might not consider how or why the megalithic might have come to adopt such a yard. I have come to trust simple integers and ratios to guide me to a possible megalithic pathway. To demonstrate, the above megalithic yard at Le Manio, of 32.625 inches is 29/32 of the English yard, and 32 lunar months (at Le Manio Quadrilateral) is 29 AMY. Such simple rationics is explored here.

By 2016 it was already obvious that the lunar month (in days) and the PMY, AMY and yard (in inches) had peculiar relationships involving the ratio 32/29, shown above. This can now be explained as a manifestation of day-inch counting and the unusual numerical properties of the solar and lunar year, when seen using day-inch counting.

It is hard to imagine that the English foot arose from any other process than day-inch counting; to resolve the excess of the solar year over the lunar year, in three years – the near-anniversary of sun and moon. This created the Proto Megalithic Yard (PMY) of 32.625 day-inches as the difference.

A strange property of N:N+1 right triangles can then transform this PMY into the English foot, when counting over a single lunar and solar year using the PMY to count months.

The metrological explanation

If one divides the three-year excess (here, the PMY) into the base then N, the normalized base of the N:N+1 triangle. In the case of the sun and moon, N is very nearly 32.625, so that the lunar to solar years are closely in the ratio 32.625:33.625. Because of this, if one counts

months instead of days,

using the three-year excess (i.e. the PMY) to stand for the lunar month,

over a single year,

the excess becomes, quite unexpectedly, the reciprocal of the PMY;

One has effectively normalized the solar year as 12.368 PMYs long. This single year difference, of 0.368 lunar months cancels with the PMY; the 0.36827 lunar months becoming 12.0147 inches. Were the true Astronomical Megalithic Yard (AMY of 32.585 inches) used, instead of the PMY, the foot of 12 inches would result. Indeed, this is the AMYs definition, as being the N (normalizing value) of 32.585 inches long, unique to the sun-moon cycle. The AMY only becomes clear, in feet, after completion of 19 solar years. This Metonic anniversary of sun and moon over 235 lunar months, is exactly 7 lunar months larger than 19 lunar years (228 months).

But this is all seen using the arithmetical methods of ancient metrology, which did not exist in the megalithic circa 4000BC. Our numeracy can divide the 1063.1 day-inches by 32.625 day-inches, to reveal the AMY as 32.585 inches long, but the megalithic could not. Any attempt to resolve the AMY in the megalithic, using a day-inch technology***, without arithmetical processes, could not resolve the AMY over 3 years as it is a mere 40 thousandths of an inch smaller than the PMY. So arithmetic provides us with an explanation, but prevents us from explaining how the megalithic came to have a value for the AMY; only visible over long itineraries requiring awkward processes to divide using factorization. However, by exploiting the coincidences of number built in to the lunar and solar years, geometry could oblige.

***One can safely assume the early megalithic resolved eighths or tenths of an inch when counting day-inches.

The geometrical explanation

In proposing the AMY was properly quantified, in the similarly early megalithic cultures of Carnac in France and the Preselis in Wales, one must turn to a geometrical method

One clue is that the yard of 3 feet (36 inches) is exactly 32/29ths of the PMY. This shows itself in the fact that 32 PMYs equal 29 yards.

Another clue is that the lunar month had been quantified (at Le Manio) by finding 32 months equalled 945 day-inches. By inference, the lunar month is therefore 945 day-inches divided by 32 or 945/32 (29.53125) day-inches – very close to our present knowledge of 29.53059 days.

From point 1, one can geometrically express any length that is 32 relative to another of 29, using the right triangle (29,32). And from point 2, since the 945 day period is 32 lunar months, as a length it will be in the ratio 29 to 32 to a length 32 PMYs long, the triangle’s hypotenuse.

Point 1 also means that 32 PMY (of 32.625 inches) will equal 1044 inches, which must also be 29 x 36 inches, and 29 yards hence handily divides the 32 side of the {29 32} right triangle into 29 portions equal to a yard on that side. One can then “mirror the right triangle about its 29-side so as to be able to draw 29 parallel lines between the two, mirrored, 32-sides, as shown in figure 1. The 945 day-inch 29-side which already equals 32 lunar months (in day-inches), now has 29 megalithic yards in that length, which are then an AMY of 945/29 day-inches!

Comparing the two AMYs and their necessary origins

Using a modern calculator, 945 divided by the PMY actually gives 28.9655 PMY and not 29, so that 945 inches requires a unit slightly smaller than the PMY and 945/29 gives the result as 32.586 inches, which length one could call the geometrical AMY. This AMY is 30625/30624 of the AMY in ancient metrology which is arrived at as 2.7 feet times 176/175 equal to 32.585142857 inches. By implication therefore, the ancient AMY is the root Drusian step whose formula is 19.008/7 feet whilst the first AMY was resolved by the megalithic to be 945/29 inches.

This geometrical AMY (gAMY?) obviously hailed from the world of day-inch counting, which preceded the ancient arithmetical metrology which was based upon fractions of the English foot. The gAMY is 32/29 of the lunar month of 29.53125 (945/32) day-inches, since 945/32 inches × 32/29 is 945/29 inches.

Using ancient metrology to interpret the earliest megalithic monuments may be questionable in the absence of a highly civilised source which had, in an even greater antiquity, provided it; from an “Atlantis”. In contrast, the monumental record of the megalithic suggests that geometrical methods were in active development and involved less sophisticated metrology, on a step-by-step basis. From this arose the English foot which, being twelve times larger than the inch, could provide the more versatile metrology of fractional feet, to provide a pre-arithmetical mechanism, to solve numerical problems through geometrical re-scaling. This foot based, fractional metrology then developed into the ancient metrology of Neal and Michell, which itself survived to become our historical metrology [Petrie and Berriman].

The two types of AMY, geometrical and the metrological, though not identical are practically indistinguishable; the AMY being just over one thousandths of an inch larger. The geometrical AMY (945/29 inches) is shown, by figure 2, to be geometrically resolvable, and so must have preceded the metrological AMY, itself only 40 thousandths of an inch less than the PMY.

The two AMYs, effectively identical, reveal a developmental history starting with day-inch counting, and division of 945 inches by 29 was made easy by exploiting the alternative factorisation of 32 PMV as 36 × 29 yards using geometry. The AMY of ancient metrology was the necessary rationalization of 945/29 inches into the foot- based system.

Bibliography for Ancient Metrology

Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.

Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.

Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.

Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.

Neal, John. All Done with Mirrors. London: Secret Academy, 2000.

—-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.

—-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.

—-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology. The Squeeze Press: 2024.

Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

It is 10 years since my brother and I surveyed this remarkable monument which demonstrates what megalithic astronomy was capable of around 4000 BC, near Carnac. The Quadrilateral is the earliest clear demonstration of day-inch counting of the solar year, and lunar year of 12 lunar months, both over three years. The lunar count was 1063.125 day-inches long and the solar 1095.75 day-inches, leaving a difference of 32.625 day-inches. This length was probably the origin of a number of later megalithic yards, which had different uses.

This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.

My thanks to Dan Palmateer of Nova Scotia for his graphics and dialogue for this series.

The last lesson showed how right triangles are at home within circles, having a diameter equal to their longest side whereupon their right angle sits upon the circumference. The two shorter sides sit upon either end of the diameter (Fig. 1a). Another approach (Fig. 1b) is to make the next longest side a radius, so creating a smaller circle in which some of the longest side is outside the circle. This arrangement forces the third side to be tangent to the radius of the new circle because of the right angle between the shorter sides. The scale of the circle is obviously larger in the second case.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

Presenting important information clearly often requires the context be shown, within a greater whole. Map makers often provide an inset, showing a larger map at a smaller scaling (as below, of South America) within a detailed map (of Southern Mexico).

Megalithic astronomy generated maps of time periods, using lines, triangles, diameters and perimeters, in which units of measure represented one day to an inch or to a foot. To quantify these periods, alignments on the horizon pointing to sun and moon events were combined with time counting between these events,where days, accumulated as feet or inches per day, form a counted length. When one period was much longer than another, the shorter could be counted in feet per day and the smaller in inches per so that both counts could share the same monumental space. In this article we find the culture leading to megalithic astronomy and stone circles, previously building circular structures called henges, made of concentric banks and ditches.