In the previous article, it was shown that the form of the trilithons, of five taller double sarsens approximating to a five-pointed star, matches the astronomical phenomena of the successive morning and evening stars, as Venus approaches Earth from the east and then recedes to the west as the morning pass. On approach, the planet rises in the evening sky and then dives into the evening sun and if one traces this motion it have the appearance of a horn. Venus shoots past the sun and reappears in the morning sky as the sun rises, creating another traced horn shape, symmetrical to the evening.
Figure 1
In support of the form of the trilithons resembling five successive double horns of Venus over 8 years, the width of the inner faces can also be interpreted as to their length being one megalithic rod, that is 2.5 megalithic yards. The distance between each pillar is ¼ of a megalithic rod so that, each inner face is divided by 4 of these units which units are 5/8ths of a megalithic yard, the ratio of the practical year of 365 days relative to the Venus synod of 584 days (365 / 584 = 5/8), the common factor between the two periods being 73 days.
The combined inner width of each pair of supports would therefore symbolize 8 x 73 days, or 584 days. Five of these pairs would then be 5 x 584 days, which equals 2920 days, this time period also being 8 x 365 days or eight practical years.
The inner surfaces lie on an ellipse which can be framed by a 5 by 8 rectangle whose sides are exactly the diameters required to form a day-counting circle for 365 day-inches (116.136 inches) and 584 day-inches (185.818 inches). The 365 day-inch circle (shown red and dashed in figure 4) has its centre in the center of the ellipse and so would have touched the two trilithons at A and B, at a tangent to their faces.
Figure 2 The Venus count of 584 day-inches (shown blue and dashed) was concentric to both the bluestone and sarsen circles, sitting inside of the bluestone circle.
The high degree of correlation between,
the five-fold form of the Venus synod and the five couplets of trilithons,
the summed inner widths of the trilithons as being 5 x 8 = 40 units of 73 days = 2920 days.
the out-rectangle of the inner ellipse being 8 by 5 and
the rectangle’s sides being the diameters of two circles of 584 inches and 365 inches, suitable for day-inch counting,
……. points to the 5-fold horseshoe of trilithons as a “temple” to the unique astronomical behavior of Venus in its synodic relationship to the solar year of 365 whole days.
No other compelling explanation exists, though many interpretions have been proposed such as
Ways of Purchasing: This large-format book, richly illustrated in color throughout, can be seen in the sidebar (on mobiles, below the tag cloud) or visit Inner Traditions.
In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its units or as a numerically different measurement. This allows all sorts of rescaling and exploitation of the properties of integer numbers.
Here we present a megalithic method which extended two or more fixed bearings (or alignments), usually based upon a simple geometrical form such as a triangle or a rectangle. This can be how the larger geometries came to be drawn on the landscape (here called landforms) of separated megaliths and natural features which appear to belong together. For example,
Outliers: Alexander Thom found that British stone circle were often associated with single outliers (standing stones) on a bearing that may correspond to horizon event but equally, appears to give clues to the metrology of the circle in the itinerary length to the outlier from the circle’s centre.
Figure 1 Stone circle plans often indicate nearby outliers and stone circles
Stone circles were also placed a significant distance and bearing away (figure 1), according to geometry or horizon events. This can be seen between Castle Rigg and Long Meg, two large flattened circles – the first Thom’s Type-A and the second his Type-B.
Figure 2 Two large megalithic circles appear linked in design and relative placement according to the geometry of the double square.
Expanding geometrically
The site plan of Castle Rigg (bottom left, fig. 1) can have the diagonal of a double square (in red) emerging between two stones which then bracket the chosen direction. This bearing could be maintained by expanding the double square so that west-to-east and south to north expand as the double and single length of a triangle while the hypotenuse then grows towards the desired spot according to a criteria such as, a latitude different to that of Castle Rigg. That is, at any expansion the eastings and northings are known as well as the distance between the two circles while the alignments, east and northeast in this example, are kept true by alignment to previous established points. Indeed, one sees that the small outlier circle of Long Meg, to Little Meg beyond, was again on the same diagonal bearing, according to the slope angle of the cardinal double square.*** One can call this a type of projective geometry.
***This extensive double square relation between megalithic sites was first developed by Howard Crowhurst, in Ireland between Newgrange and Douth in same orientation as figure 2, and by Robin Heath at https://robinheath.info/the-english-lake-district-stone-circles/.
It seems impossible for such arrangements to have been achieved without modern equipment and so the preference is to call these landforms co-incidental. But, by embracing their intentionality, one can see a natural order between Castle Rigg and, only then, Long Meg’s outlying Little Meg circle, and through this find otherwise hidden evidence of the working methods in the form of erratics or outliers, whose purpose is otherwise unclear.
Equilateral Expansion
The work of Robin Heath in West Wales can be an interesting challenge since not all the key points on his Preseli Vesica are clearly megalithic, perhaps because megaliths can be displaced by settlements or be subsumed by churches, castles and so on. (see Bluestone Magic, chapter 8). First, for completeness, how is a vesica defined today? In his classic Sacred Geometry, Robert Lawlor explains the usual construction and properties of the vesica :
Drawing 2.3. Geometric proof of the √3 proportion within the Vesica Piscis. from Sacred Geometry by Robert Lawlor.
Draw the major and minor axes CD and AB. Draw CA, AD, DB and BC. By swinging arcs of our given radius from either centre A or B we trace along the vesica to points C and D, thus verifying that lines AB, BC, CA, BD and AD are equal to one another and to the radius common to both circles.
We now have two identical equilateral triangles emerging from within the Vesica Piscis. Extend lines CA and CB to intersect circles A and B at points G and F. Lines CG and CF are diameters of the two circles and thus twice the length of any of the sides of the triangles ABC and ABD. Draw FG passing through point D.
Sacred Geometry by Robert Lawlor
Primitive versus later geometry
Lawlor’s presentation have the triangles appearing as the conjuction of two circles and their centers. However, the points and lines of modern geometry translate, when interpreting the megalithic, into built structures or significant features, and the alignments which may join them. The alignments are environmental and in the sky or landscape.
A is Pentre Ifan, a dolmen dating from around 3500 BC.
B is located in the Carningli Hillfort, a mess of boulders below the peak Carningli (meaning angel mountain). Directly East,
C is the ancient village, church and castle of Nevern.
D is a recently excavated stone circle, third largest in Britain at around 360 feet diameter, but now ruinous, call Waun Mawn.
The two equilateral triangles have an average side length around 11,760 feet but, as drawn, each line is an alignment of azimuth 330, 0, 30, and 90 degrees and their antipodes.
The Constructional Order
Relevant here is how one would lay out such a large landform and we will illustrate how this would be done using the method of expansion.
North can be deduced from the extreme elongation of the circumpolar stars in the north, since no pole star existed in 3200BC. At the same time it is possible to align to plus and minus 30 degrees using Ursa Major. This would give the geometry without the geometry so to speak, since ropes 11760 feet long are unfeasible. It seems likely that the Waun Mawn could function as a circumpolar observatory (as appears the case at Le Menec in Brittany, see my Lords of Time).
If the work was to start at Carningli fort, then the two alignments (a) east and (b) to Waun Mawn could be expanded in tandem until the sides were 11760 feet long, ending at the circle to the south and dolmen to the east. The third side between these sites should then be correct.
Figure 3 Proposed use of equilateral expansion from Carnigli fort to both what would become the dolmen of Pentre Ifan (az. 90 degrees) and Waun Mawn (azimuth 150 degrees).
The vesica has been formed to run alongside the mountain. The new eastern point is a dolmen that points north to another dolmen Llech-y-Drybedd on the raised horizon, itself a waypoint to Bardsey Island.
The reason for building the vesica appears wrapped up in the fact that its alignments are only three, tightly held within a fan of 60 degrees pointing north and back to the south. But the building of the double equilateral cannot be assumed to be related to the circular means of its construction given by Lawlor above. That is, megalithic geometry did not have the same roots as sacred geometry which has evolved over millennia since.
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.
Perched on a hill in the forest north of the Carnac alignments, a megalithic site has escaped the fences that have littered the landscapes of the region for several years. These are the menhir and the quadrilateral of Manio. From the outset, the large menhir impresses with its dimensions. Nearly 5m50 high, it is the highest standing stone in the town.
More discreet, the quadrilateral caps the top. 90 upright and contiguous stones, varying in height between 10 cm and 1m60, make up an enclosure approximately 36 meters long and 8 meters wide on average, because the long sides converge. The stones at the ends draw a curve. Four stones to the northeast form the remains of a circle. Two menhirs, much larger than all the other stones in the quadrilateral, open a kind of door in the south file. This particular form questions us. What could she be used for? Was it a meeting place, maybe an enclosure for sheep? In fact, what we see today is probably only the outer skeleton of a larger monument, a mound of stone and earth that contained a chamber inside. Other remains complicate the whole, unless they help us solve our puzzle. Hidden in the brambles and brush, we can discover a stone on the ground of rounded shape. These curves are reminiscent of the belly of a pregnant woman. She is nicknamed the “Lady” of the Manio.
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.
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.
