Vectors in Prehistory 1

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.

At Le Manio’s southern curb, the excess of the solar year over the lunar year, over 3 years, is 32.625 (32 and 5/8ths) day-inches, which is probably the first of many megalithic yards of around 2.72 feet, then developed for specific purposes (Appendix 2 of Language of the Angels). At Le Manio, the solar year count was shown above the southern curb, east of the “sun gate”, but many other counts were recorded within that curb, as a recording of many lengths, though the lunar year was the primary baseline and the 14 degree sightline above the curb aligned to the summer solstice sunrise.

Numbers-as-symbols, and arithmetic, did not exist. Instead, numbers-as-lengths, of constant units such as the inch, were generated as measurements of different types of year. To know a length without our numeric system required the finding of how a given number of units divided into a length, in an attempt to know the measurement through its observed factorization. This habit of factorization could start with the megalithic yard itself as having been naturally created from the sky, as Time. In this case, when the megalithic yard was divided into the three lunar year count of 1063.1 days, the result was 10.875 (10 and 7/8th) “times” 32.625 day-inches. which is one third of the megalithic yard, and is the number of day-inches of the excess for a single solar year.

The lunar year is the combined result of lunar motion, in its orbit, and solar motion along the ecliptic, of average of one day-in-angle per solar day. The lunar year is the completion of twelve cycles of the moon’s phases. The counting at Le Manio hinged upon the fact that, in three solar years there was a near-anniversary of 37.1 lunar months. This allowed the excess to be very close to the invariant form of the solar-lunar triangle which can be glimpsed for us by multiplying the lunar month (29.53059 days) by 32/29 to give 32.58548. (see also these posts tagged 32/29).

The excess of the solar year, in duration and hence in measured length, the 0.368 (7/19) lunar months (over 12), almost exactly equals the reciprocal of the megalithic yard (19/7 feet) so that, when lunar months are counted using megalithic yards, the excess becomes 12 inches which is 32/29 of 10.875 day-inches. From this it seems likely that the English foot and megalithic yard were generated, as naturally significant units, when day-inch counting was applied to the solar and lunar years.

The Manio Quadrilateral may have been like a textbook, a monumental expression of Megalithic understanding, originally built over the original site of that work or, carried from a different place in living memory. It presents all manner of powerful achievements, such as the accurate approximation of the lunar month as 29.53125 (945/32) days, the significance of the eclipse year, alignment to the solstice maximum and lunar minimum standstill, the whole number count over 4 years of 1461 days – then available as a model of the ecliptic, and a circular Octon simulator – and much else besides. This megalithic period preceded the English stone-circle culture initiated by Stonehenge 1 around 3000 BC but was somewhat contemporary with the Irish cairn and dolmen building culture. Metrology is presented near Carnac as a work-in-progress, based closely on astronomy rather than land measurement as such.

My work on the Megalithic tools-and-techniques can be read in my Lords of Time and in Language of the Angels, further considered as a tradition inherited by ancient world monumentalism. This post will be followed soon by more on vectors in prehistory.

The Integration of the Megalithic Yard

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.

My 2012 Post below discusses John Neal’s view of the megalithic yard
drawing on his ancient metrology.

John Neal makes a masterful job of considering the megalithic yard in the context of historical metrology, a metrology that he has managed to forge into a single conceptual scheme in which measures known to history from different lands all inter-relate.

Neal’s book, All Done With Mirrors, is one of the most fundamental and significant contributions to the late megalithic and ancient world understanding of numbers but to read it is no easy matter since he takes no prisoners and fully expects readers to resolve through calculation what he does not explicitly state. This makes his approach different to mine in which I try to present as easily a possible aids to the visualisation and registration of a pattern of facts. However, neither approach can really substitute for what one has to do for oneself in order to understand and John gave his “Secret Academy” idea the catch line “We can’t give it away” because of the often deafening silence with which his work is met.

The aim here is to give some workings based on Neal’s book, to give others a taste of what lies beneath what is written and also to further my own interests in the Megalithic Yard. Thom’s lack of metrological background led to both an original approach but also a disconnect to what is known about historical metrology. One particular mystery is how measures appear to have propagated unchanged across millennia.

Neal says on page 47:

Thom made a comparison of his Megalithic Yard with only one other known unit of measurement. This was the Spanish vara, the pre-metric measurement of Iberia, its value 2.7425 feet. Related measurements to the vara survive all over the Americas wherever the Spanish settled, from Peru to Texas. Although the vara is exactly one of the lengths of the m.y. the fact that it is divided into three feet makes this relationship uncertain. These feet are thought to be Roman but this belief is also unlikely, and they would appear to be related to the earlier Etruscan-Mycenaean units. This is a good example of an intermediate measure being thought to be related because of a similarity in length, and illustrates the importance of considering the sub-divisions when sourcing a measure.

How units of measure are divided and aggregated follows strict rules. If these rules did not exist then the system of metrology would have no inner structure as a system. We don’t expect measures to follow rules because today we simply measure things, and do everything else as a calculation following on from that. Metrology is an “ology” because it is a system of calculation that was used for building ancient structures when only limited calculation was possible.

Thus Neal can talk about the ancestry of the megalithic yard because the forensic tools are available through the system of metrology, in which a yard has three feet but that places the foot at close to the limits for a foot, at just over 0.9 feet, for the vara which would then be a yard of near Assyrian feet (9/10 feet). The Roman foot is far greater at 24/25 or 0.96 feet. A Mycenean foot would be 15/16 of the Roman which is in the region of 0.91 feet but the compounding to two errors, that the vara is a yard and that the Roman is its foot is the sort of confusion that only an exact metrology can ever recover from.

Neal continues:

Why he [Thom] did not analyze the Megalithic Yard in terms of what was already very well known of ancient metrology, must remain a mystery. And why, after the Megalithic Yard becoming the most scrutinized measure in the history of measure, nobody else has succeeded in doing so, is an even greater mystery. The very simple fact of the matter is, that if as Thom claimed from the beginning, the Megalithic Yard has 40 sub-divisions, then it is not a “yard” but a double remen [1.25], or 2 and 1/2 feet, and the “megalithic inch” is a digit! If the Megalithic Yard is taken to be 2.7272 feet, which is within Thom’s parameters for the value, the megalithic inch is .06818 feet, which is well within the range of the digits of all known ancient measurements. 16 of these digits are therefore one megalithic foot of 1.0909 English feet. This is a well-known measurement in ancient metrology, sometimes referred to as the Ptolemaic foot, and mistakenly, as the Drusian foot. His “fathom” of 2 m.y. is the historically well-known intermediate measurement, of a pace of 5 feet. Then, his “megalithic rod” [6.8 feet] is 6.25 Ptolemaic feet, which is also a measure known in antiquity as being 100th part of a furlong of 625ft or 1/8th part of the 5,000ft mile. The megalithic measures are not, therefore, peculiar to what is accepted as the megalithic arena, but are perfectly integrated with measuring systems found throughout the ancient world.

One should realize here that Neal is using the word “ancient” in an unquantified way because he believes metrology and other sciences of the numerical arts were inherited by the megalithic – a position that I question since there is no evidence for it. The megalithic could have generated a science of metrology in its earliest phase which then evolved into the greater system of many types of feet (Neal’s modules) since the older megalithic monuments have not been well studied – the British monuments being from a later phase. The early burial mounds, if found to have employed this fuller system, would prove Neal’s thesis. he continues,

Furthermore, the methods whereby Thom discovered [his megalithic measures], namely by careful surveys and comparisons, are the time honoured methods pioneered by Petrie and in no way are they a mistaken interpretation of the evidence, or invention.

The pattern of metrology comes in the ratios between types of unit. If a different foot is used these patterns remain constant and when metrology is used to analyse monuments then it this grammar of its usage that has remained invariant. This may seem to be geeky nonsense until metrology is resolved as a system within which the apparent babel of metrological signals become a direct communication from the past. Neal does not make this any easier by delivering a masterly analysis that prerequires most of the structural understandings to be in place.

Doth this profit a man? And is it simply a specialist field? For sure, by now, like Neal I am something of a specialist. It is true that no older language than metrology, other than language itself, has come down from such antiquity. If there is a truth behind claims (like mine) that the number sciences were sacred and contain mysteries concerning the spiritual world, metrology could be a philosopher’s stone. But when and how?

It is also true that this system of prehistoric thought is now a very powerful forensic tool for recovering their intended meaning of ancient sites and the types of measure found might reveal lines of metrological transmission in the ancient world. Anyone interested needs to apply it in practice.

This excerpt was first published on in 2012


The only problem in adopting Neal’s full structure for ancient metrology is that it bears upon the type of metrological knowledge of the size and shape of the Earth, that lies behind the form of the Great Pyramid and other ancient buildings. But I have since seen, from the point of view of early megalithic astronomy, a much freer use of the ordinal numbers {1 2 3 4 5 6 7 8 9… etc} was applied to counts of astronomical time, using simple geometries of circles and right triangles within which a simpler metrology arose, as explained in Sacred Geometry: Language of the Angels. Another problem with Neal’s metrological grid of “Earth ratios” is that the modular range becomes so filled with versions of each foot that a given measurement can give one a false identification upon which a false interpretation or dead end can defeat the process.

This means that, earlier than the late megalithic, one is studying primitive ratios between astronomical measurements. This is clear at Crucuno Dolmen to Rectangle, where the month was coded as 27 feet but the day was the root Iberian foot of 32/35 feet. From this can be deduced an accurate approximation of the lunar month as 27 feet divided by 32 and multiplied by 35 giving 29.53 125 (27 x 35/ 32) Iberian feet. When one multiplies this month by 32 (the denominator) the result is 945 so that 945 days equals 32 lunar months. It is therefore true that the original three lunar year count (leading to the megalithic yard) is 36 months, two lunar years 24 months and two Jupiter synods are 27 lunar months. This forms a limiting octave of {18 24 27 36} which became Plato’s World Soul in his Timaeus cosmogony 6:8::9:12 only tripled [do1 fa sol do2} (see my Harmonic Origins of the World). From this the megalithic can be seen to naturally lead finding 27 lunar months between three loops of Jupiter, so that one Jupiter synod is 13.5 (27/2) months. Hence my reconstruction of the Pythagorean Music of the Spheres, as a mystery garnered from the megalithic.

Geometry 6: the Geometrical AMY

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.

Figure 1 The three solar year count’s geometrical demonstration of the excess in length of 3 solar years over 3 lunar years as the 32.625 day-inch PMY.

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

  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!

Figure The 29:32 relationship of the PMY to the yard as 32 PMY = 29 yards whilst 32 lunar months (945 days) is 29 AMY.

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

  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. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  4. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  5. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016.
  6. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  7. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

Day-inch counting at the Manio Quadrilateral

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.

Continue reading “Day-inch counting at the Manio Quadrilateral”

Geometry 5: Easy application of numerical ratios

above: Le Manio Quadrilateral

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. diagram of Dan Palmateer.

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

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Models of Time within Henges and Circles

image: composite, see figure 1 below

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).

This map is shown in the context of South America with a yellow rectangle which is the part blown up in scale. The subject is the Quetzal birds range which corresponds well to the Olmec then Maya heartlands leading to the god named Quetzalcoatl or Feathered Serpent. (see chapter 8 of Heath, 2018.)

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.

Thornborough Henge
Continue reading “Models of Time within Henges and Circles”