Author: Richard Heath

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.

Wikipedia

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.

Continue reading “The Megalithic Numberspace”

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.

Counting Perimeters

above: a slide from my lecture at Megalithomania in 2015

We know that some paleolithic marks counted in days the moon’s illuminations, which over two cycles equal 59 day-marks. This paved the way for the megalithic monuments that studied the stars by pointing to the sky on the horizon; at the sun and moon rising to the east and setting in the west. It was natural then to them to see the 12 lunar months (6 x 59 = 354 day-marks) within the seasonal year (about 1/3 of a month longer than 12) between successive high summers or high winters.

Lunar eclipses only occur between full moons and so they fitted perfectly the counting of the repetitions of the lunar eclipses as following a fixed pattern, around six months apart (actually 5.869 months, ideally 173.3 day-marks apart). The accuracy of successive eclipse seasons to the lunar month can then improve over longer counts so that, after 47 lunar months, one can expect an eclipse to have occurred about one and a half days earlier. This appears to be the reason for the distance between the megalithic monuments of Crucuno, its dolmen and and its rectangle, which enabled simultaneous counting of days as Iberian feet and months as 27 foot units, at the very end of the Stone Age.

Continue reading “Counting Perimeters”

The Best Eclipse Cycle

The anniversary of the Octon (4 eclipse years in 47 lunar months) did not provide similar eclipses and so, by counting more than four, the other motions of the Moon could also form part of that anniversary. This is especially true of the anomalistic month, which changes the changes the apparent size of the Moon within its phase cycle, recreate the same type of lunar eclipse after nineteen eclipse years. This 18 year and 11 day period is now taken as the prime periodicity for understanding eclipse cycles, called the Saros period – known to the Babylonian . The earliest discovered historical record of what is known as the saros is by Chaldean (neo-Babylonian) astronomers in the last several centuries BC.

The number of full moons between lunar eclipses must be an integer number, and in 19 eclipse years there are a more accurate 223 lunar months than with the 47 of the Octon. This adds up to 6585.3 days but the counting of full moon’s is obviously ideal as yielding near-integer numbers of months.

We noted in a past post that the anomalistic month (or AM), regulating the moon’s size at full moon, has a geometrical relationship with eclipse year (or EY) in that: 4 AM x pi (of 3.1448) equals the 346.62 days of the eclipse year as the circumference. Therefore, in 19 EY the diameter of a circle of circumference 19 x 346.62 days must be 4 x 19 AM so that , 76 AM x pi equals 223 lunar months, while the number of AM in 223 lunar months must be 239; both 223 and 239 being prime numbers.

Continue reading “The Best Eclipse Cycle”

Vectors in Prehistory 2

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.

Continue reading “Vectors in Prehistory 2”