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.

How Geometries transformed Time Counts into Circles

Above: example of the geometry that can generate one or more circles,
equal to a linear time count, in the counting units explained below.

It is clear, one so-called “sacred” geometry was in fact a completely pragmatic method in which the fourfold nature of astronomical day and month counts allowed the circularization of counts, once made, and also the transmission of radius ropes able to make metrological metrological circles in other places, without repeating the counting process. This “Equal Perimeter” geometry (see also this tag list) could be applied to any linear time count, through dividing it by pi = 22/7, using the geometry itself. This would lead to a square and a circle, each having a perimeter equal to the linear day count, in whatever units.

And in two previous posts (this one and that one) it was known that orbital cycles tend towards fourfold-ness. We now know this is because orbits are dynamic systems where potential and kinetic energy are cycled by deform the orbit from circular into an ellipse. Once an orbit is elliptical, the distance from the gravitational centre will express potential energy and the orbital speed of say, the Moon, will express the kinetic energy but the total amount of each energy combined will remain constant, unless disturbed from outside.

In the megalithic, the primary example of a fourfold geometry governs the duration of the lunar year and solar year, as found at Le Manio Quadrilateral survey (2010) and predicted (1998) by Robin Heath in his Lunation Triangle with base equal to 12 lunar months and the third side one quarter of that. Three divides into 12 to give 4 equal unit-squares and the triangle can then be seen as doubled within a four-square rectangle, as two contraflow triangles where the hypotenuse now a diagonal of the rectangle.

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The Octon of 4 Eclipse Years

Having seen, in the last post, that three eclipse years fitted into the three-year count at Le Manio, another eclipse fact has come to light, recorded within the nearby site of Crucuno, between its dolmen and rectangle. The coding of time at Crucuno was an evolution of a new metrology based upon the English foot in which, the right triangle of longest integer side lengths was replaced by fractions of a foot using the same two numbers as the sides would have had. This allowed the measurement of a time period to be simultaneously seen in both days and months. That this was possible can be seen at Le Manio, where it could be noticed that 32 lunar months equaled exactly 945 day-inches.

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The Quantification of Eclipse Cycles

Following on from the last post:
Given the many sub-cycles found in the Moon’s behavior, and the angle of its orbit to the Ecliptic, one would expect the eclipse phenomenon to be erratic or random but in fact eclipses repeat quite reliably over relatively fixed periods that were quantified symbolically by megalithic astronomy, within monuments and by the “sacred” numbers and geometries which encapsulate eclipse cycles, as with many other cycles.

An eclipse cycle repeats, to greater or lesser degree of accuracy, over an integer number of days or months. And because of a lack of conventional arithmetic or notation like our own in the megalithic, the practical representation of a cycle would be a raw count of days or months, using uniform measures, which could then be interpreted by them using (a) the rational fractions of whole unit metrology, (b) the factorization of a measured length by counting within using measuring rods or (c) using right-triangles or half-rectangles, which naturally present trigonometrical ratios; to compare different time cycles.

The Eclipse Year

The solar year (365.242 days) is longer than the lunar year of 12 lunar months (354.367 days) and we know that these, when counted in day-inches, gave the megalithic their yard of 32.625 (32 and 5/8) inches and that, by counting months in megalithic yards over one year, the English foot (of 12 inches) was instead the excess over a single lunar year of the solar year, of 12.368 lunar months. 0.368 in our notation is 7/19 and the megalithic yard is close to 19/7 feet so that counting in months cancels the fraction to leave one foot.

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The Fourfold Nature of Eclipses

The previous post ended with a sacred geometrical diagram expressing the eclipse year as circumference and four anomalous months as its diameter. The circle itself showed an out-square of side length 4, a number which then divides the square into sixteen. If the diameter of the circle is 4 units then the circumference must be 4 times π (pi) implying that the eclipse year has fallen into a relationship with the anomalous month, defined by the moon’s distance but visually by manifest in the size of the moon’s disc – from the point of view of the naked eye astronomy of the megalithic.

In this article I want to share an interesting and likely way in which this relationship could have been reconciled using the primary geometry of π, that is the equal perimeter model of a square and a circle, in which an inner circle of 11 units has an out-square whose perimeter is, when pi is 22/7, 44.

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The Strange Design of Eclipses

We all know about solar eclipses but they are rarely seen, since the shadow of the moon (at one of its two orbital nodes) creates a cone of darkness which only covers a small part of the earth’s surface which travels from west to east, taking hours. For the megalithic to have pinned their knowledge of eclipses to solar eclipses, they would have instead studied the more commonly seen eclipse (again at a node), the lunar eclipse which occurs when the earth stands between the sun and the moon and the large shadow of the earth envelopes a large portion of the moon’s surface, as the moon passes through our planet’s shadow.

This phenomenon of eclipses is the result of many co-incidences:

Firstly, if the orbit of the moon ran along the ecliptic: there would be a solar eclipse and a lunar eclipse in each of its orbits, which are 27 and 1/3 days long.

Secondly, if the moon’s orbit was longer or shorter, the angular size of the sun would not be very similar. The moon’s orbit is not circular but elliptical so that, at different points in the lunar orbit the moon is larger, at other points smaller in angular size than the sun. This is most visible with solar eclipses where some are full or total eclipses, and others eclipse less than the whole solar disc, called annular eclipses.

Thirdly, the ecliptic shape of the moon’s orbit is deformed by gravitational forces such as the bulge of the earth, the sun and planets so that its major axis rotates. When the moon is furthest away (at apogee), its disc exceeds that of the sun. And when the moon is nearest to the earth (at perigee), its disc is smaller than that of the sun. This type of progression is called the precession of the lunar orbit where the major axis travels in the same direction as the sun and moon. This contrasts with the precession of the lunar nodes which also rotate (see later).

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