Geometry 6: the Geometrical AMY

By 2016 it was already obvious that the lunar month (in days) and the PMYproto-megalithic yard of 32.625 (261/8) day-inches, generated at Le Manio Quadrilateral as the difference between three solar and three lunar year counts., AMYA megalithic yard which, in inches, expresses the true astronomical ratio of mean solar months to lunar months. and yard (in inches) had peculiar relationships involving the ratio 32/29Ratio between the astronomical megalithic yard (AMY) as 945/29 and the lunar month of 29.53125 as 945/32 day-inches., shown above. This can now be explained as a manifestation of day-inch countingThe practice of counting the days, using inches or other small units, between synodic phenomena such as years or planetary loops. 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 footThe standard prehistoric foot (of 12 inches) representing a unity from which all other foot measures came to be formed, as rational fractions of the foot, a fact hidden within our historical metrology [Neal, 2000]. arose from any other process than day-inch counting; to resolve the excess of the solar yearFrom Earth: the time in which the sun moves once around the Zodiac, now known to be caused by the orbital period of the Earth around the Sun. over the lunar year, in three years – the near-anniversary of sun and moon. This created the Proto Megalithic YardAny unit of length 2.7-2.73 feet long, after Alexander Thom discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. (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 32.625:33This is the number of years for an exact number of 12053 days. This period can be measured using the equinoctal sun and it has come to be known as the lifetime of semi-divine Solar Heroes such as Jesus and Mithras. This period relates geometrically to the 18.618 years of the moon's nodal period..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 MetonicGreek: The continuous 19 year recurrence of the moon's phase and location amongst the stars. 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 metrologyThe application of units of length to problems of measurement, design, comparison or calculation., 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 CarnacAn extensive megalithic complex in southern Brittany, western France, predating the British megalithic. in France and the Preselis in Wales, one must turn to a geometrical method

1. 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.
2. 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 176/175Ratio crucial to maintaining integers (see geometry lesson 2) between radii and circumference of a circle, and crucial to the micro-variation of foot modules in ancient metrology. 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

1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
2. 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.
3. Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
4. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
5. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
6. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pior π: The constant ratio of a circle's circumference to its diameter, approximately equal to 3.14159, in ancient times approximated by rational approximations such as 22/7. and the World.
7. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
8. —-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology.  The Squeeze Press: 2024.
9. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.