From Sacred Geometry: Language of the Angels 2

Part two (part one) of a serialisation from Sacred Geometry: Language of the Angels, Appendix 1.
(book available in first few weeks of 2021)
This is relevant to many on-site posts.

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

From Sacred Geometry: Language of the Angels

Part one of a serialisation 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.

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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.

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.

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

Continue reading “Geometry 5: Easy application of numerical ratios”

Geometry 4: Right Triangles within Circles

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.

This lesson is a necessary prequel to the next lesson.

It is an initially strange fact that all the possible right triangles will fit within a half circle when the hypotenuse equals the half-circles diameter. The right angle will then exactly touch the circumference. From this we can see visually that the trigonometrical relationships, normally defined relative to the ratios of a right triangle’s sides, conform to the properties of a circle.

A triangle with sides {3 4 5} demonstrates the general fact that, when a right triangle’s hypotenuse is the diameter of a circle, the right angle touches the circumference.
Continue reading “Geometry 4: Right Triangles within Circles”

Geometry 3: Making a circle from a counted length

The number of days in four years is a whole number of 1461 days if one approximates the solar year to 365¼ days. This number is found across the Le Manio Quadrilateral (point N to J) using a small counting unit, the “day-inch”, exactly the same length as the present day inch. It is an important reuse of a four-year count to be able to draw a circle of 1461 days so that this period of four years can become a ouroboros snake that eats its own tale because then, counting can be continuous beyond 1461 days. This number also permits the solar year to be counted in quarter days; modelling the sun’s motion within the Zodiac by shifting a sun marker four inches every day.

Figure 1 How a square of side length 11 will equal the perimeter of a circle of diameter 14

Our goal then is to draw a circle that is 1461 day-inches in perimeter. From Diagram 1 we know that a rope of 1461 inches could be divided into 4 equal parts to form a square and from that, an in-circle to that square has a diameter equal to a solar year of 365¼ days. Also, with reference to Figure 1, we know that the out-circle will have a diameter of 14 units long relative to the in-circle diameter being 11 units long, and this out-circle will have the perimeter of 1461 inches that we seek.

Figure 3 A general method, using the equal perimeters model, applied to a 4 solar year day count of 1461 day-inches, found as a linear count at the Manio Quadrilateral. A square, formed from this linear count, can be transformed into an outer circle of equal perimeter using the simple geometry of π as 22/7.

For this, the solar year rope (the in-circle diameter) needs to be divided into 11 parts. Start by choosing a number that, when multiplied by 11, is less that 365 (and a 1/4). For instance, 33. A new rope will be formed, 11 x 33 = 363 inches, marked every 33 inches to provide 11 divisions. Through experience, we discover we need 2 identical ropes so as to make practical use of the properties of symmetry through attaching ropes to both ends of the solar diameter rope.

Place one rope at the West side of the in-circle diameter and swing it up until it touches the in-circle. Place the other rope at the East side of the in-circle diameter and swing it down until it touches the edge of the in-circle. Now connect the 33 inch marks between the 2 ropes. This will divide the 365 1/4 diameter into 11 segments.

Seven of those segments are the new radius to create the 1461 inch outer-circle.

Figure 3 Division of the in-circle into eleven equal parts so as to select 7 units as a radius rope to then form the circle of diameter 14 units and perimeter 1461 inches.

This novel application of the equal perimeters model, rescued from Victorian textbooks by John Michell and applied by him most memorably perhaps to Stonehenge and the Great Pyramid (in Dimensions of Paradise) is a general method for taking a counted length and reliably forming a radius rope able to transform that counted length into a circle of the same perimeter as the square, easily formed by four sides ¼ of the desired length.

The site survey at the start, drawn by Robin Heath, appeared in our survey of Le Manio.

A Pyramidion for the Great Pyramid

image: By 1200 BC, the end of the Bronze Age, the Egyptian map of the world (above) showed nine bows or latitudes, numbers 4 to 9 including the Nile Delta, Delphi, Southern Britain and Iceland, a map based on an ancient geodetic survey.

This post explores a pyramidion, now lost, which exceeded the apex height of the pyramid, so as to model the different reference latitudes established by geodetic surveys and encoded within their metrology and the Great Pyramid (by 2500 BC). This pyramidion would have sat on the flat top of the pyramid, 480 feet above the base of the pyramid.

In All Done With Mirrors, John Neal described how the full height of the pyramid, reaching to its natural apex, would have been just over 481 feet. Most pyramids probably had a pyramidion since a number have been found elsewhere that repeat aspects of or have a name carved on them, of a specific pyramid. Sitting on their apex, they often repeat the form of the larger pyramid, and are scale models of a specific pyramid. In the case of the Great Pyramid, exactly 441th of its natural apex is missing, and this is likely to be because a pyramidion once stood on the flat top the actual pyramid.

Continue reading “A Pyramidion for the Great Pyramid”