Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

The Megalithic Pythagoras

Pythagoras of Samos (c.600BC) very likely gleaned megalithic number science on his travels around the “Mysteries” of the ancient world. His father, operating from the island of Samos, became a rich merchant, trading by sea and naming his child Pythagoras; after the god of Delphi who had “killed” the Python snake beneath Delphi’s oracular chasm, now a place of Apollo. The eventual disciples of Pythagoras were reclusive and secretive, threatening death on anybody who would openly speak of mysteries, such as the square root of two, to the uninitiated. It can be seen from the previous post that many such “mysteries” were natural discoveries made by the megalithic astronomers, when learning how to manipulate number without arithmetic, through a metrological geometry unfamiliar to the romantic sacred geometry of “straight edge and compass”.

As previously stated, the vertex angles of right triangles whose longer sides are integer in length, are angular invariants belonging to the invariant ratio of their sides. To create a {11 14} angle one can use any multiple of 11 and the same multiple of 14 to obtain the invariant angle whereupon, the hypotenuse and base will shrink or grow together in that ratio: any length on the “14” line is 14/11 of any length below it on the “11” base line and visa versa.

If one enlarges the base line to being 99 then the diagonal of the square side length 99 will be 140, which is 99 times the square root of two. In choosing, as I did, to enlarge 91 (the quarter year) to 9 x 11 = 99, I encountered the cubit of the Samian (“of Samos”) foot of 33/35 feet, as follows. When Heraclitus, also of Samos, visited the Great Pyramid he gave its southerly side length as 800 “of our feet” and 756 English feet (the measured length) needs to be divided by 189 and multiplied by 200 to obtain such a measurement, giving a Samian foot of 189/200 (=0.945 feet) which is 441/440 of the Samian root foot of 33/35 feet. 33/35 x 3/2 = 99/70 (1.4143) feet but its inverse of 35/33 x 4/3 = 140/99 feet.

There is then no doubt about Samos as being a center in the Greek Mysteries since, the form of the Greek temple seems first to evolve there. For example, 10,000 feet of 0.945 feet equal 945 feet, the number of days in 32 lunar months. The Heraion of Samos (pictured above) has been shown to have had pillars around a platform (a peristyle), and an elongated rectangular room (a cella), involving megalithic yards and a 4-square geometry cunningly linking lunar and solar years, to alignments to the Moon’s minimum using the {5 12 13} second Pythagorean Triangle. (diagram at top is from figure 5.9 of Sacred Geometry: Language of the Angels).

The reason for the Samian (lit. “of Samos”) foot being 33/35 feet appears to be that as a cubit of 99/70 feet, or √2 =1.4142, it is the twin of 140/99 as 1.41. In the geometrical world such foot ratios were exact, relative to the English foot; which is the root of the Greek module and of all other rational modules, such as the Royal of 8/7 feet. Such cubits could measure across the diagonal the same number as the side length in English feet. Such measures became essential for building of rectangular temple structures in Greece and further east, but when the metrological geometry, of square and circle in equal perimeter, was the focus, 140 in the diagonal can use 99 in the base (or side-length of the square).

If we remember that the 99 length must be rooted from the shared center of the square and equal circle then, the side length of the square must be twice that, or 198. This means that the perimeter of the square must be 4 times that, equal to 792, at which point readers of John Michell’s books on models of the world will recall that the diameter of the mean earth can be presented, within an equal perimeter design, if each unit is multiplied by 720 units of 10 miles, my own summary being in my recent Sacred Geometry book , chapter 3 on measuring the Earth. This model Michell called The Cosmological Prototype, where the mean earth diameter is (quite accurately) 7920 miles.

If the square of 198 feet is rolled out into a single line, it “becomes” the mean diameter of the Earth in units of 10 miles. For this sort of reason, my 2020 book was called Language of the Angels, since this model looks like a first approximation of the mean earth size which a later Ancient Metrology would improve upon as to accuracy, by a couple of miles! That is, that the earth’s dimensions follow a design based upon metrological geometry and the properties of numbers.

John Michell finalized his Cosmological Model in an Appendix to The Sacred Center, and in his text on “sacred Geometry, Ancient Science, and the Heavenly Order on Earth” called The Dimensions of Paradise, both published by Inner Traditions.

Seven, Eleven and Equal Perimeters

above: image of applications involving sacred geometry based upon pi as 22/7 and a circle of equal perimeter to a square, from a previous post.

The geometrical and other relationships between different numbers are easily found to be useful through simple experiments. The earliest approximations to pi (22/7) was key in the megalithic and later ancient cultures, for making circles of a known diameter and circumference, the foremost using the numbers 7 and 11 doubled twice. A staked rope of length seven will create a circumference of 44, to a high degree of accuracy.

But what is pi? it actually connects two different worlds, of extensive linear measure and of intensive rotational measure. As the radius rope is made larger the circle expands from its center but it remains a whole circle, except that its circumference is made up of more “units” all according to the ratio pi = 22/7, in a good approximation.

But measuring a circumference is fiddly, it is circular! In contrast, it is very much easier to work with squares since their perimeter is four times their side length. And in many cases, one does not really need to measure the perimeter. Because of this, the megalithic looked for and discovered an easier procedure in which one could know the circumference of a circle if one could generate the square that has the same circumference now called the equal perimeter model. This was surprisingly simple to grasp and implement.

First of all, one can lay out a linear length, that divides by 4, lets say 28 which is 4 x 7. The length is made up of four lengths, each of 7 units and, a square of side length 7 will have a perimeter of 28, same as the linear length. The square is really just a rolled-up set of 4 lengths at right angles!

The diameter of a circle with 28 units on its circumference must be larger than its incircle of diameter 7 and, if pi is 22/7 then, the diameter will be exactly 14/11 of the side length. Notice that 14/11 is cancelling the seven and eleven in pi as 22/7.

The equal perimeter rope will be staked in the very center of the square. The side of 7 is then 7 x 14/11 or 98/11 units and this, times 22/7 equals 28 – the perimeter of both the circle, and square side-length 7 units. There is no need to calculate this if one draws a triangle ratio {11 14} from the center of the square. This triangle’s slope angle automatically “calculates” or reproportions the cardinal length (whatever this is) into a suitable rope (or radiant) length.

One often does not need to form the circle to know what its perimeter would be through measurement. Once one knows that every square has a twin circle of the same perimeter, this changes thinking. This is particularly significant when forming a circular model of the sun’s path in the year. If the “saturnian” year 364 days was used, it unusually divides by 28 days, and 13, and 7 days; the seven-day week. The square would have a side length of 13 weeks (91 days) and the radius rope would need to be (13 x 7) x 7/11 which, times 44/7 reconstitutes the circumference of 364 days.

My book Sacred Geometry: Language of the Angels has much to say on equal perimeter modelling, which is found throughout the ancient building traditions that followed on from the megalithic period, using the older techniques of metrological geometry alongside the development of arithmetic methods. Click on the Bookshop logo or Google, and find out more.

The Megalithic Numberspace

above: counting 37 lunar months six times to reach 222,
one month short of 223: the strong Saros eclipse period.

There is an interesting relationship between the multiple interpretations of a number as to its meaning, and the modern concept of namespace. In a namespace, one declares a space in which no two names will be identical and therefore each name is unique and this has to be so that, in computer namespaces such as web domain names, the routes to a domain can be variable but the destination needs to be a unique URL.

If sacred numbers had unique meanings then they would be like a namespace. Instead, being far more limited in variety, sacred numbers have more meanings, or interpretations, just as one might say that London has many linkages to other cities. In an ordinal number set, there are many relationships of a number to all the other numbers. This means whilst their are infinite numbers in the set of positive whole numbers, there are more than an infinity of relationships between the members of that set, such as shared number factors or squares, cubes, etc. of a number.

The mathematician Georg Cantor saw “doubly infinite” sets. Sets of relationships between members of an already infinite set, must themselves be more than infinite. He called infinite sets as aleph-zero and the sets of relationships within an infinite set (worlds of networking), he called aleph-one.

Originally, Cantor’s theory of transfinite numbers was regarded as counter-intuitive – even shocking.


However, in the world of sacred numbers, although there can be large numbers, in the megalithic the numbers were quite small; partly due to the difficulty that numbers-as-lengths were physically real while later numeracy abstracted numbers into symbols and, using powers of ten, modern integers are a series of place ordered numbers (not factors) in base 10, as with 12,960,000 – possible for the ancient Babylonians but, I believe, not expected for the early megalithic.

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The Knowing of Time by the Megalithic

The human viewpoint is from the day being lived through and, as weeks and months pass, the larger phenomenon of the year moves the sun in the sky causing seasons. Time to us is stored as a calendar or year diary, and the human present moment conceives of a whole week, a whole month or a whole year. Initially, the stone age had a very rudimentary calendar, the early megalith builders counting the moon over two months as taking around 59 days, giving them the beginning of an astronomy based upon time events on the horizon, at the rising or setting of the moon or sun. Having counted time, only then could formerly unnoticed facts start to emerge, for example the variation of (a) sun rise and setting in the year on the horizon (b) the similar variations in moon rise and set over many years, (c) the geocentric periods of the planets between oppositions to the sun, and (d) the regularity between the periods when eclipses take place. These were the major types of time measured by megalithic astronomy.

The categories of astronomical time most visible to the megalithic were also four-fold as: 1. the day, 2. the month, 3. the year, and 4. cycles longer than the year (long counts).

The day therefore became the first megalithic counter, and there is evidence that the inch was the first unit of length ever used to count days.

In the stone age the month was counted using a tally of uneven strokes or signs, sometimes representing the lunar phase as a symbol, on a bone or stone, and without using a constant unit of measure to represent the day.

Once the tally ran on, into one or more lunar or solar years, then the problem of what numbers were would become central as was, how to read numbers within a length. The innovation of a standard inch (or digit) large numbers, such as the solar year of 365 days, became storable on a non-elastic rope that could then be further studied.

The 365 days in he solar year was daunting, but counting months in pairs, as 59 day-inch lengths of rope, allowed the astronomers to more easily visualize six of these ropes end-to-end, leaving a bit left over, on the solar year rope, of 10 to 11 days. Another way to look at the year would then be as 12 full months and a fraction of a month. This new way of seeing months was crucial in seeing the year of 365 days as also, a smaller number of about 12 and one third months.

Twelve “moons” lie within the solar year, plus some days.

And this is where it would have become obvious that, one third of a month in one year adds up, visually, to a full month after three years. This was the beginning of their numerical thinking, or rationality, based upon counting lengths of time; and this involved all the four types of time:

  1. the day to count,
  2. the month length to reduce the number of days in the day count,
  3. the solar year as something which leaves a fraction of a month over and finally,
  4. the visual insight that three of those fractions will become a whole month after three full solar years, that is, within a long count greater than the year.

To help one understand this form of astronomy, these four types of time can be organized using the systematic structure called a tetrad, to show how the activity of megalithic astronomy was an organization of will around these four types of time.

J.G. Bennett’s version of Aristotle’s tetrad.

The vertical pair of terms gives the context for astronomical time on a rotating planet, the GROUND of night and a day, for which there is a sky with visible planetary cycles which only the tetrad can reveal as the GOAL. The horizontal pair of terms make it possible to comprehend the cosmic patterns of time through the mediation of the lunar month (the INSTRUMENT), created by a combination of the lunar orbit illuminated by the Sun during the year, which gave DIRECTION. Arguably, a stone age culture could never have studied astronomical time without Moon and Sun offering this early aggregate unit of the month, then enabling insights of long periods, longer than the solar year.

The author (in 2010) at Le Manio Quadrilateral
where megalithic day-inch counting is clearly indicated after a theodolite survey,
over three years of its southern curb (to the left) of 36-37 stones.

The Manio Quadrilateral near Carnac demonstrates day-inch counting so well that it may itself have been a teaching object or “stone textbook” for the megalithic culture there, since it must have been an oral culture with no writing or numeracy like our own. After more than a decade, the case for this and many further megalithic innovations, in how they could calculate using rational fractions of a foot, allowed my latest book to attempt a first historical account of megalithic influences upon later history including sacred building design and the use of numbers as sacred within ancient literature.

The “output” of the solar count over three years is seen at the Manio Quadrilateral as a new aggregate measure called the Megalithic Yard (MY) of 32.625 (“32 and five eighths”), the solar excess over three lunar years (of 36 months). Repeating the count using the new MY unit, to count in months-per-megalithic yard, gave a longer excess of three feet (36 inches), so that the excess of the solar year over the lunar could then be known as a new unit in the history of the world, exactly one English foot. It was probably the creation of the English foot, that became the root of metrology throughout the ancient and historical world, up until the present.

The southern curb (bottom) used stones to loosely represent months from point P while, in inches, the distance to point Q’ was three solar years.

This theme will be continued in this way to explore how the long counts of Sun, Moon, and Planets, were resolved by the megalithic once this activity of counting was applied, the story told in my latest book.