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 proceeded 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.

From Sacred Geometry: Language of the Angels

from Sacred Geometry: Language of the Angels, Appendix 1.
(Available: first few weeks of 2021)
This is relevant to many on-site posts.

Metrology has appeared in modern times (phase five below) in reverse order, since humankind saw the recent appearance of many measures in different countries as indicative that past cultures made up units of measure as and when they needed them, perhaps based upon lengths found in the human body. But this soon breaks down under scrutiny because the measures called after different regions all have systematic ratios between them, such as 24/25 feet (which as a foot is the Roman) and 6/5 feet (which is an aggregate unit, a remen), and the size of humans is quite various between regions and within populations. As stated in the main body of this book, the notion of measures from different regions was called historical metrology. This framework began to break down when answers appeared as to why the different regional feet were related, not only to the English foot as equalling one for each ratio, but also to the fact that the units of measure were often seen to divide into the size and shape of the Earth (leading to our phase four)—then called ancient metrology.

Another aspect of measures was their ability to approximate important, otherwise irrational, constants (our phase 3), such as π, √2 and even e in the form of megalithic yards, which are close to 2.71828 feet, the numerical value of e—the exponential constant. The earliest megalithic yard was almost exactly that number of feet—derived from an astronomical count over three lunar and solar years in day-inches (chapter 1) leaving a 32.625-inch difference between these years (our phase one); those 32.625 inches equal 2.71875 (87/32) feet.

The gap between the first and second phases of metrology seems to be the gap in time between the megalithic in Brittany and in Britain. Only as the metrological purpose of more megalithic monuments becomes clear might one be able to know more accurately, but British metrology, in choosing a megalithic yard of 2.72, was able to factor the nodal prime number of 17 within its counting. While Brittany could, at Le Ménec’s western cromlech, use a radius of 17 megalithic rods (6.8 feet) to have a count of 3400 megalithic inches across a diameter, Britain could use 12 such rods to model the lunar year of 12 months while also counting 15 rods as 3400 shu.si, a small digit known to historical metrology as dividing the 1.8 foot (the double Assyrian foot of 0.9 feet) into 60 parts, while the shu.si (0.03 feet) divides into many foot modules (see p. 112), and the English yard contains 100 shu.si, and 68 yards contains 6800 shu.si enabling the nodal period to be counted at Balnuaran in Scotland.

There is a particular need to regularize this subject through the gathering of more examples of metrology’s past applications. One must recognize that those responsible for our present knowledge of it have largely passed away, and those in academia are not going to rewrite history in order to impartially reassess whether their own approach to ignoring it can still be justified, especially when they are not preserving the metrology within monuments because they can’t see it as a signal from the past.

Overview of Megalithic Units of Measure

At least five specific MYs have emerged from the counting applications within megalithic monuments:

1. The proto megalithic yard (PMY) of 32.625 day-inches, emanating from an original day-inch count over 3 solar and 3 lunar years (at the Manio Quadrilateral) as the difference in their duration (chapter 1). This is therefore an artifact of the world of inch counting.

2. The Crucuno megalithic yard (CMY) of 2.7 feet: We saw that, by the factorization of 32 lunar months as 945 days long, the lunar month (as 29.53125 days long) can be represented by 10 MYs of 2.7 feet (27 ft) where the days in such a count are the Iberian foot of 32/35 feet. This I call the Crucuno megalithic yard, though, in the historical period, this foot came to be called the root foot (27/25 feet) of the Drusian module, which, times 25, is then 27 feet. The astronomical megalithic yard AMY (next) is 176/175 of the CMY.

3. The astronomical megalithic yard (AMY): In Britain, this is 2.715 feet (32.585 inches) long, giving N = 32.585 for the actual N:N + 1 differential ratio between the solar and lunar years. When representing lunar months over a single year, the excess becomes the English foot of 12 inches—a megalithic, now-called English, foot. From this one sees that every AMY on the base of the Lunation Triangle defines an AMY plus 1 inch on the hypotenuse above it (length N + 1 = 33.585 inches – a Spanish vara), as the duration 1 mean solar month. The AMY can appear as an integer when the CMY defines a radius because it is 176/175 of the CMY.

4. The nodal megalithic yard (NMY): Used in Britain. Thom’s Megalithic Sites in Britain gave the megalithic yard as having had the value of 2.72 feet as “the” MY, based on integer geometries within stone circles and some statistical methods applied to some of the other inter-stone distances Thom had measured. Its value evidently derives from its relationship to the nodal period of 6800 day-feet because 2.72 =6800/2500, where 2500 feet is half a metrological mile of 5000 feet. For this reason, I now call it the nodal megalithic yard (NMY), which contains the key prime number 17 in its formula 272/100, 272 being 16 times 17. Its megalithic rod (NMY times 2.5) of 6.8 feet factorized the nodal period of 6800 days: 15 rods gave 102 feet (3400 shu.si) and 30 rods gave 204 feet (6800 shu.si – e.g. Clava and Avebury), the shu.si being 204/6800 = 3/100 feet. It therefore appears that the NMY, its rod of 6.8 feet, and the shu.si had a raison d’être in the British megalithic period that was focused on the later problem in astronomy of counting the days of the nodal period.

5. The later* megalithic yard (LMY): Seen at Stonehenge and Avebury. Thom in 1978 published a new estimate for the MY as 2.722 feet. Unbeknownst to Thom but lurking within his own error bars was a further development of the AMY which, times 441/440, would locate his value within ancient metrology as 2.716 feet, 126/125 of the CMY. The CMY is clearly the root value (in Neal’s terminology 2.5 root Drusian of 27/25 feet) and the AMY the root canonical value, while this LMY is the standard canonical value.
*in the context of Thom’s work.

All of these different megalithic yards had their place in the megalithic people’s pursuit of their astronomical knowledge. Noting the role of the shu.si in compressing the length of a nodal count to a mere 204 feet, Thom’s NMY of 2.72 is the key to how its length of 3/100 feet was arrived at. The shu.si of 0.03 feet (0.36 inches) surprisingly divides into many of the historical modules of foot-based metrology.

Historical
Module
Foot Ratioshi.siNotes
Assyrian 9/1030Carrying the sexagesimal (base-60)
system of the Sumerians.
Roman42/2532
Inverse Byzantine99/10033Times 3 gives 99, a yard minus one shu.si.
English133.3Times 3 gives 100 shu.si in a yard.
?51/5034Divides into the nodal period.
The difference between 80 and
81.6 feet and between 90 and
91.8 feet at Seascale, where 91.8
locates the Jupiter synodic period.
Persian21/2035Its remen (6/5) is 42 shu.si.
Drusian27/2536The CMY is root of the AMY and
the LMY.
Remen6/540Half-remen of 20 shu.si as ideal
form of the equal perimeter model.
Some units commensurate with the shu.si

Five Phases for Metrology

MetrologyThe application of units of length to problems of measurement, design, comparison or calculation. as a single system was based on the number 1, which was then realized astronomically as 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]. as an excess over one year [Robin Heath, 1998; Heath & Heath 2010], which then became related to all the foot modules of the ancient world—through a range of simple fractions. There were, therefore, phases in the evolution of ancient and then historical
metrology. I can see five right away.

Phase One: An Inch-Based Metrology for Astronomical Counting*

Primordial measures arising from the conduct of astronomy in the megalithic period included the English inch used to count days at Le Manio, CarnacAn extensive megalithic complex in southern Brittany, western France, predating the British megalithic.; 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. (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.) of 261/8 inches arising from Le Manio’s three-year count, forming the Lunation TriangleThe right-angled triangle within which the lengths of the two longer sides are the relative proportions of the solar and lunar years.; and the English foot arising from counting the Lunation Triangle over a single solar year as lunar months using the PMY per month.

To form the English foot required definite steps that were necessarily taken
through megalithic astronomy and findable in the monumental record as (a) the use of the inch to count days over 3 solar years,† (b) the use of the differential length over 3 years to count lunar months rather than days, and (c) the counting over a single year to find an excess length of the English foot, which still has 12 inches because the lunar year has 12 months.

*Corresponding to the work of Heath and Heath (2011) and Heath (2014)
†See “Reading the Angelic Mind” in chapter 1, p. 14.

Phase Two: A Foot-Based Metrology for Astronomical Counting‡


Using ad hoc simple foot ratios based upon the English foot, in the service of
astronomical counting such as: 27 feet representing the lunar month at Crucuno (near Carnac) enabling days to be counted in parallel, using Iberian feet of 32/35 feet; nodal units such as Thom’s early megalithic yard of 2.72 feet; and the yard of 3 feet containing 100 shu.si.§ Nearby, the use of feet per day can be seen at Erdevan, over the SarosThe dominant eclipse period of 223 lunar months after which a near identical lunar or solar eclipse will occur. and MetonicGreek: The continuous 19 year recurrence of the moon’s phase and location amongst the stars. periods. The full system of Ancient Metrology was not yet developed.

‡Corresponding to the work of Alexander Thom (1967, 1971, 1978, 1980)
§More on the types of megalithic yard and the shu.si can be found in the box above (p. 237)
called “Overview of Megalithic Units of Measure.”

Phase Three: A Foot-Based Metrology for Handling Mathematical Functions

Using ratios of the English foot to approximate to irrational and geometric functions: measures are able to map feet to √2 or its reciprocal, to π, or to other measures related to the models in chapter 2.

The English foot was long enough to form fractional ratios in which the number field could be expressed as a calculating tool, since the measurement of a length using a different ratio of foot length gives a result in which the original measurement has been multiplied by the denominator of the fraction and divided by the numerator. Thus, 9 feet becomes 8 feet of the ratio 9/8. The initial approach to such ratio-based feet was to build right triangles using English feet so that the foot of 9/8 feet emerged from a base length of 8 feet and hypotenuse of 9 feet. Above each foot on the base were 8 demarcated feet of 9/8 feet (see fig. 2.1, p. 34), and there are strong reasons to suspect grids of unit squares were in use to form triangles since the right angle is native to such grids, which are also conceptually adapted to studying pure numerical interactions in space.

Phase Four: A Metrology of Foot-Based Modules and Microvariations*

Foot modules evolved as a general-purpose toolkit involving only the prime numbers {2 3 5 7 11}: the systems of root measures using right triangular ratios from the common English foot standard; a common grid of microvariation within each module, applicable to geodeticUnits of measures and monumental measurements relating to the numerical definition of the shape of the Earth by the late megalithic. surveying and modeling; and some less common microvariations such as 225/224 and 81/80.

*Corresponding to the recent books about ancient metrology
from John Michell (1981, 2008) and John Neal (2000, 2016, 2017)

Phase Five: The Foot-Based Metrology Discovered from the Historical Period†

The historical measures were found through exploration of the geographical regions after which they were named, such as measuring sticks, anthropomorphic sculptures, objects whose size was noted in antiquity, modern-era survey measurements (e.g., by Petrie and Thom), and through inductive metrology, measuring surviving sites and artifacts.

†Corresponding to Petrie (1877) and Berriman (1953)

BIBLIOGRAPHY

  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.

Recalibrating the Pyramid of Giza

Once the actual height (480 feet) and actual southern base length (756 feet) are multiplied, the length of the 11th degree of latitude (Ethiopia) emerges, in English feet, as 362880 feet. However, in the numeracy of the 3rd millennium BC, a regular number would be used. In the last post, it was noted that John Neal’s discovery of such rectangular numbers to define degrees of latitude, multiplied the pyramid’s pointed height (481.09 feet) by the southern base length (756 feet) to achieve the length of the Nile Delta degree of latitude and, repeating Neal’s diagram relating the key latitudinal degrees of the ancient Model as figure 1, the Ethiopian degree is 440/441 of the Nile Delta degree. As shown above, the length of the 756 foot southern base is changed, when re-measured in the latitudinal feet for Ethiopia; it becomes the harmonic limit of 720 feet of 1.05 feet – normally called the root Persian foot.

Continue reading “Recalibrating the Pyramid of Giza”

Ethiopia within the Great Pyramid

My last posting mentioned John Neal’s creative step of not averaging the Great Pyramid of Giza’s four sides, as had routinely been done in the past – as if to discover an idealized design with four equal sides. Instead, Neal found each length to have intensionally been different. When multiplied by the pyramid’s full height, the length of four different degrees of latitude were each encoded as an area. The length of the southern side is integer as 756 feet, and this referred to the longest latitude, that of the Nile Delta, below 31.5 degrees North. Here we find that the pyramid’s reduced height also indicated the latitude of Ethiopia.

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Geometry 2: Maintaining integers using fractions

understanding the megalithic: circular structures: part 2

The megalithic sought integer lengths because they lacked the arithmetic of later millennia. So how did they deal with numbers? There is plenty of evidence in their early monuments that today’s inch and foot already existed and that these, and other units of measure, were used to count days or months. From this, numbers came to be known by their length in inches and later on as feet, and longer lengths like a fathom of five feet, the cubit of 3/2 feet and, larger still, furlongs and miles – to name only a few.

So megalithic numeracy was primarily associated with lengths, a system we call metrology. Having metrology but not arithmetic, the integer solutions to problems became a necessity. Incidentally, it was because of their metrological numeracy that the megalithic chanced upon a rich seam of astronomical meaning within the geocentric time world that surrounds us, a seam well-nigh invisible to modern science. Their storing of numbers as lengths also led to their application to the properties geometrical structures have, to replicate what arithmetic and trigonometry do, by using right triangles and a system of fractional measures of a foot (see later lesson – to come). In what follows, for both simplicity and veracity, we assume that π was too abstract for the megalithic, since they first used radius ropes to create circles, so that 2π was a more likely entity for them to have resolved.

Continue reading “Geometry 2: Maintaining integers using fractions”

Units within the Great Pyramid of Giza

There is a great way to express pi of 22/7 using two concentric circles of diameter 11 and 14 (in any units). Normally, a diameter of 7 gives rise to a circumference of 22, when pi is being approximated as 22/7 (3.142587) rather than being the irrational number 3.141592654 … for then, the 14 diameter should have a circumference of 44, which is also the perimeter of the square which encloses a circle of diameter 11.

The square of side 11 and
the circle of diameter 14
will both have the same perimeter.

Figure 1 The Equal Perimeter model of two circles, the smaller of which has an out-square of equal perimeter to the greater circle
Continue reading “Units within the Great Pyramid of Giza”