Ad Quadratum is a convenient and profound technique in which continuous scaling of size can be given to square shapes, either from a centre or periphery. The differences in scale are multiples of the square root

of two [sqrt(2)] between two types of square: cardinal (flat) and diamond (pointed).

## Thornborough Henge as Moon’s Maximum Standstill

The three henges appear to align to the three notable manifestations to the north west of the northerly moon setting at maximum standstill. The distance between northern and southern henge entrances could count 3400 days, each 5/8th of a foot (7.5 inches), enabling a “there and back again” counting of the 6800 days (18.618 solar years/ 19.618 eclipse years) between lunar maximum standstills.

Continue reading “Thornborough Henge as Moon’s Maximum Standstill”## The Tetraktys as plan of planetary harmony and the four Elements

In "Planetary Resonances with the Moon" I explored the astronomical matrix presented in *The Harmonic Origins of the World* with a view to reducing the harmonic between outer planets and the lunar year to a single harmonic register of Pythagorean fifths. This became possible when the 32 lunar month period was realized to be exactly 945 days but then that this, by the nature of Ernest McClain’s harmonic mountains (figure 1) must be 5/4 of two Saturn synods.

Using the lowest limit of 18 lunar months, the commensurability of the lunar year (12) with Saturn (12.8) and Jupiter (13.5) was “cleared” using tenths of a month, revealing Plato’s World Soul register of 6:8::9:12 but shifted just a fifth to 9:12::13.5:18, perhaps revealing why the Olmec and later Maya employed an 18 month “supplementary” calendar after some of their long counts.

By doubling the limit from 18 to three lunar years (36) the 13.5 is cleared to the 27 lunar months of two Jupiter synods, the lunar year must be doubled (24) and the 32 lunar month period is naturally within the register of figure 1 whilst 5/2 Saturn synods (2.5) must also complete in that period of 32 lunar months.

Continue reading “The Tetraktys as plan of planetary harmony and the four Elements”## Number Symbolism at Table des Marchands

Table des Marchands, a dolmen at Lochmariaquer, can explain how the Megalithic came to factorise 945 days as 32 lunar months by looking at the properties of the numbers three, four and five. At that latitude, the solstice angle of the sun on the horizon shone along the 5-side of a 3-4-5 triangle to east and west, seen clearly at the Crucuno Rectangle "Lunar Counting from Crucuno Dolmen to its Rectangle".

Before numbers were individually notated (as with our 3, 4 and 5
rather than |||, |||| and |||||) and given positional notation (like our
decimal seen in 945 and 27), numbers were lengths or marks and, when marks are
compared to accurately measured lengths measured out in inches, feet, yards,
etc. then *each vertical mark would naturally
have represented a single unit of length*. This has not been appreciated
as having been behind marks like the cuneiform for ONE; that it probably meant
“one unit of length”.

## Lunar Counting from Crucuno Dolmen to its Rectangle

It is not immediately obvious the Crucuno dolmen (figure 1) faces the Crucuno rectangle about 1100 feet to the east. The role of dolmen appears to be to mark the beginning of a count. At Carnac’s Alignments there are large cromlechs initiating and terminating the stone rows which, more explicitly, appear like counts. The only (surviving) intermediate stone lies 216 feet from the dolmen, within a garden and hard-up to another building, as with the dolmen (see figure 2). This length is interesting since it is twice the longest inner dimension of the Crucuno rectangle, implying that lessons learned in interpreting the rectangle might usefully apply when interpreting the distance at which this outlier was placed from the dolmen. Most obviously, the rectangle is 4 x 27 feet wide and so the outlier is 8 x 27 feet from the dolmen.

Continue reading “Lunar Counting from Crucuno Dolmen to its Rectangle”## Earth and Moon within Westminster’s Coronation Pavement

Our modern globes are based upon political boundaries and geographical topography yet they have geometrical predecessors, which described the world as an image, diagram or schemata. The original idea for the form of the world was summarised within a simple two dimensional geometry, like an eastern mandala or yantra.

Such a diagram was built into the Cosmati pavement at Westminster Abbey, built by Henry III and dedicated to the Saxon King and Saint Edward the Confessor. This exotic pavement became the focus for the Coronations of subsequent English then British monarchs.

Continue reading “Earth and Moon within Westminster’s Coronation Pavement”## Musical Tones of the Outer Planets

My crucial entré to planetary harmony came when I noticed musical ratios in the synodic time periods of Jupiter and Saturn relative to the lunar year. This approach differs from the norms for “harmonies of the spheres” (a.k.a. Musica Universalis) which are geometrical and spatial, rather than temporally harmonic.

The planetary harmony I found within synodic periods became the subject of my new book *The Harmonic Origins of the World (pub. *2018). These synodic ratios have been parts of my work from c. 2000, then expressed as “matrix diagrams” (*Matrix of Creation, *figure 2 below). In my new book, I show how **ancient tuning theory** seems to have presented the same information, in a different type of matrix (see figure 4).

Below I connect the outer planets using two additional (and useful) kinds of diagram, the right-angled triangle (figure 1) and the Pentad (figure 5), the latter developed in the 20th century within a discipline called *Systematics. *

## REVIEW of The Harmonic Origins of the World

During the latter part of the twentieth century, three divergent speculative perspectives opened up on the ancients’ cosmology: astronomical, musical, and metrological. The astronomical perspective found its classic expression in von Dechend and de Santillana’s *Hamlet’s Mill*. The musical perspective was spelled out, almost single-handedly, by Ernest McClain, beginning with his work *The Myth of Invariance*. The metrological perspective diverged into the practical (descending from Alexander Thom’s surveying in the nineteenth century), and the more theoretical work associated perhaps most famously with John Michell’s *View Over Atlantis*.

These three perspectives shared an awareness that *number* was an
indispensable guide. Number is invariant; three is always three, and always one
plus two. Mathematics is a realm of order, and recurrent patterns like the
seasons or the harmonic scale call for mathematical descriptions precisely
because such descriptions find stability in change.

As scoffers and skeptics like to point out, however, where there is
pattern-finding, there is also often unconscious wishful human ingenuity.
Moreover, because the astronomical, musical, and metrological perspectives were
carried on sometimes in isolation from one another, their results diverged, and
an apparent incommensurability emerged: how could they *all* be true? This
gave scoffers an argument that was, on the face of it, difficult to answer: why
not *none* of them instead? Perhaps the real answer was the skeptical
shrug: the ancient myth-tellers and builders of stone circles were acting more
or less haphazardly or moved by very terrestrial, local, and historical
concerns. Was this not the *simplest* explanation?

Richard Heath for a quarter of a century has been building towards a
case diametrically opposed to this. From the beginning he worked with
Thom’s practical metrological results, bringing them into dialogue with Michell
and John Neal; then later with a further expansion of astronomical results that
far outpaced von Dechend and de Santillana’s speculations on the precession of
equinoxes. In *The Harmonic
Origins of the World*, Heath goes a further step,
bringing McClain’s results into dialogue with his previous work. Heath
provides ample demonstration that the results of these various
perspectives can clearly be seen to *not* diverge from one another. Suddenly
it is very plausible that they might indeed “all be true,” because
they were never, for the ancients, separate at all.

According to Heath, there exists in our solar system a harmony of extraordinary beauty among planetary cycles. This harmony was observed by ancient astronomers, and enshrined in megalithic monuments; it was transmitted in oral and literary culture via a musical grammar of proportion, easily reproducible across various cultures, which informs scripture and speculation (in McClain’s phrase) “from the Rg Veda to Plato.” These assertions are of course controversial and deserve scrutiny. But they give the lie to any facile dismissal of ancient cosmological sophistication on the grounds that reconstructions are inconsistent. Astronomy, metrology (practical and theoretical), and music are all comprehensible under a single analogical system. They hang together in a coherent, living dialogue.

This book is the most recent chapter and the most comprehensive introduction to a vital adventure in ideas. It is a detailed account of how human beings on the ground could make sense of the sky by way of the octave. In it, rigor and common sense meet wonder and awe.

## Models of Time within Henges and Circles

Presenting important information clearly often requires the context be shown, within a greater whole. Map makers often provide an inset, showing a larger map at a ** smaller scaling** (as below, of South America) within a detailed map (of Southern Mexico).

Megalithic astronomy generated maps of time periods, using lines, triangles, diameters and perimeters, in which units of measure represented one day to an inch or to a foot. To quantify these periods, **alignments **on the horizon pointing to sun and moon events were combined with **time counting** between these events,where days, accumulated as feet or inches per day, form a counted length. When one period was much longer than another, the shorter could be counted in feet per day and the smaller in inches per so that both counts could share the same monumental space. In this article we find the culture leading to megalithic astronomy and *stone *circles, previously building circular structures called **henges**, made of concentric banks and ditches.

## Iceland’s Model of the Earth’s Meridian

Einar Palsson [1, at end] saw that the myths of foundation for Iceland’s settlement in 930 had Pythagorean roots. Since then Petur Halldorsson has identified patterns that could not have been influenced by Pythagoras (c. 600 BC) and Pythagoras was known to have adapted the existing number sciences found (according to his myth) from Egypt to China.

Such patterns, called Cosmic Images by Halldorsson [3], seek to establish a geometric connection between places on the landscape and on the horizon, here in the south-western region near Reykjavik, the only Icelandic city. The spirit of a region or island was integrated through organising space in this way, according to centers (Things) of circles and their radius and diameter as numbers of paces, circles punctuated with places and alignments to other places, horizon events or cardinal directions. John Michell provided a guide to some of the techniques in his books [2, at end].

Continue reading “Iceland’s Model of the Earth’s Meridian”