Wikipedia diagram by David Eppstein : This is an updated text from 2002, called “Finding the Perfect Ruler”
Any number with limited “significant digits” can be and should be expressed as a product of positive and negative powers of the prime numbers that make it up. For example, 23.413 and 234130 can both be expressed as an integer, 23413, multiplied or divided by powers of ten.
What Primes are
Primes are unique and any number must be prime itself or be the product of more than one prime. Having no factors, prime numbers are odd and cannot be even since the number 2 creates all the even numbers, meaning half of the ordinals are not prime once two, the first “number” as such, emerges.
Each number can divide one (or any other number) into that number of parts. In the case of three (fraction 1/3) only one in three higher ordinal numbers (every third after three) will have three in it and hence yield an integer when three divides it.
Four is the first repetition of two (fraction ½) but also the first square number, which introduces the first compound number, the geometry of squares and the notion of area.
Ancient World Maths and Written Language
The products of 2 and 3 give 6, 12, etc., and the perfect sexagesimal like 60, 360 were combined with 2 and 5, i.e. 10, to create the base 60, with 59 symbols and early ancient arithmetic, in the bronze age that followed the megalithic and Neolithic periods.
In early education of applied mathematics, there was a simple introduction to vector addition: It was observed that a distance and direction travelled followed by another (different) distance and direction, shown as a diagram as if on a map, as directly connected, revealed a different distance “as the crow would fly” and the direction from the start.
The question could then be posed as “How far would the plane (or ship) be, from the start, at the end”. This practical addition applies to any continuous medium, yet the reason why took centuries to fully understand using algebraic math, but the presence of vectors within megalithic counted structures did not require knowledge of why vectors within geometries like the right triangle, were able to apply vectors to their astronomical counts.
A previous post explained the anatomy of the primary celestial cycles of the Sun and Moon. The “resting” part of these cycles are the winter solstice (opposite the summer solstice which was today) and the dark moon (which is coming in a week, after the waning half moon day before yesterday). In the resting phase, the cosmological origin is traditionally found, containing all that is to manifest but that is not yet expressed. In this respect, the Big Bang is the equivalent for modern thinking, as the origin of the entire visible and invisible universe seen via modern instrumentation and discoveries.
Life is somehow connected with our large Moon, without which there could have been no living planet. The form of life appears influenced by the moon and its conjunctions with different planets. And without (a) the tides, (b) the tectonic plates supporting continents, and (c) the tilt and spin of the earth; the earth would be static rather than actively supporting the necessary rhythms of Life. A primordial collision created these features of our earth and moon, since the cyclic archetypes provide an essential framework for living beings, to which their bodies are synchronized through circadian and behavioral rhythms.
above: a 21-petal object in the Heraklion Museum which could represent the 21 seven-day weeks in the 399 days of the Jupiter synod. [2004, Richard Heath]
One of the unfortunate aspects of adopting the number 360 for calibrating the Ecliptic in degrees is that the megalithic counted time in days and instead saw the ecliptic as divided by the 365¼ days. In transferring to the number 360, with all of its easy factors, 8 x 9 x 5, moderns cannot exploit a key advantage of 365¼ days.
If the lunar orbit takes 27.32166 days then each day the moon moves by 1/27.32166 of the ecliptic every day. For this reason, after 27.32166 days the orbit completes because the Moon’s “year” then equals one as the angular motion has been 27.32166/ 27.32166 = 1.
The same is true of the lunar nodes, which retrograde to the east along the ecliptic in 18.618 years. For this reason one can say, the lunar nodes move by 1/18.618 DAYS (in angle) every day and to travel one DAY in angle, the nodes take 18.618 DAYS per day (needing the new term “node day” equal the 18.618 days.*** A solar year takes 19.618 node days (since 365¼ equals 18.618 x 19.618) and an eclipse year takes 18.618 x 18.618 – 346.62 days
*** These are average figures since the moon comes under variable gravitational influences that are episodic.
A general rule emerges in which the larger, whole cycles, lead to reciprocals which can be numerically characterized by knowing the number of the days in the larger period.
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.
