Author: Richard Heath

π and the Megalithic Yard

The surveyor of megalithic monuments in Britain, Alexander Thom (1894 – 1985), thought the builders had a single measure called the Megalithic Yard which he found in the geometry of the monuments when these were based upon whole number geometries such as Pythagorean triangles. His first estimate was around 2.72 feet and his second and final was around 2.722 feet. I have found the two megalithic yards were sometimes 2.72 feet because the formula for 272/100 = 2.72 involved the prime number 17 as 8 x 17/ 100, and this enabled the lunar nodal period of 6800 days to be modelled and and the 33 year “solar hero” periods to be modelled, since these periods both involve the prime number 17 as a factor. In contrast, Thom’s final megalithic yard almost certainly conformed to ancient metrology within the Drusian module in which 2.7 feet times 126/125 equals 2.7216 feet, this within Thom’s error bars for his 2.722 feet as larger than 2.72 feet.

This suggests Thom was sampling more than one megalithic yard in different regions or employed for different uses. Neal [2000] found for Tom’s statistical data set having peaks corresponding to the steps of different modules and variations in ancient metrology, such as the Iberian with root 32/35 feet and the Sumerian with root 12/11 feet. It is only when you countenance the presence of prime numbers within metrological units that one breaks free of the inevitably weak state of proof as to what ancient units of measure actually were and, more importantly, why they were the exact values they were and further, how they came to be varied within their modules. However, the megalithic yard of 2.72 appears to outside the system in embodying the prime number 17 for the specific purpose of counting longer term periods which themselves embody that prime number.

The discipline of using only the first five primes {2 3 5 7 11} must have been accompanied by the perception that only if primes were dealt with could certain ends be served. This is crystal clear when we come to musical ratios in which the harmonic primes alone are used of {2 3 5} with an occasional “passenger” of the prime {7} as in 5040 which is 7 x 720, the harmonic constant.

Using 2.72 feet to count the Nodal Period

The first remarkable characteristic of 2.72 feet is that 8 x 17 in the numerator means that the approximation to π of 25/8 = 3.125 can, in (conceptually) multiplying a diameter, provide an image of 25 units on the circumference of a stone circle. For example a diameter of 2 MY would suggest 17 MY on the circumference, which is quite remarkable. Further to this, we know that the 6800 days of nodal cycle is factored as 17 x 400 and that the MY was shown (acceptably) to have been made up of 40 digits (in conformance to the general tradition within metrology that there are 16 digits per foot and 40 for a step of 2.5 feet, which a MY traditionally is). The circumference of 17 MY is then 17 x 40 digits which means that a diameter of 20 MY would give a circumference of 17 x 400 digits equalling 6800 digits as a countable circumference in digits per day.

This highlights how prime number factors played a role, in the absence of arithmetical methods, in solving the astronomical problems faced by the late stone age when counting time periods in days.

The Broch that Modelled the Earth

Summary

In the picture above [1] the inner profile of the thick-walled Iron-Age broch of Dun Torceill is the only elliptical example, almost every other broch having a circular inner court. Torceill’s essential data was reported by Euan MacKie in 1977 [2]: The inner chamber of the broch is an ellipse with axes nearly 23:25 (and not 14:15). The actual ratio directly generates a metrological difference, between the major and minor axis lengths, of 63/20 feet. When multiplied by the broch’s 40-foot major axis, this π-like yard creates a length of 126 feet which, multiplied again by π as 22/7, generates 396 feet. If each of these feet represented ten miles, this number is an accurate approximation to the mean radius of the Earth, were it a sphere.

The two ratios involved, 22/7 and 63/20, each an approximation to π, become 9.9 (99/100) when they are multiplied together, as an approximation to π squared.  Figure 1 shows that these two ratios, if 22/7 differently used as its reciprocal 7/22, also generates the ratio between the mean and polar radii of the Earth, since 63/20 x 7/22 = 441/440. The ancient Meridian length could be calculated from 396 when multiplied by using the most accurate rational π noted by Fibonacci as 864/275. The 396 units, of 10 miles per foot, was a practical distance to have realized in the megalithic without arithmetic, to store the 3960 mile mean radius of the earth, since the mile of 5280 feet is 4/3 of 3960; that is, 396 x 4/3 equals 528, implying that this model was conceived of within a decimal framework but without the base-10 positional notation of arithmetic. We show that the methods of calculation used can only have seen numbers-as-lengths as being composed of factors of just the first five prime numbers {2 3 5 7 11} and that this limitation upon numbers created a metrology in which fractional units of measure could manipulate lengths to multiply and divide them through addition and subtraction of the powers of these primes.

Marc Calhoun’s picture from the Island (picture from his blog)

Contents

  1. Summary
  2. Introduction.
  3. Main Thesis.
  4. Pre-arithmetic Calculation using Powers of {2 3 5 7 11}
  5. Combining Prime Number Composites.
  6. Appendix 1 Extract from Science and Society in Prehistoric Britain.
  7. Appendix 2: Preface: The Metrology of the Brochs.
  8. Metrological Bibliography.
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Mangroves and the Moon’s Maximum

photo: Ariefrahman for Wikipedia /_

Tides on earth are due to the sun and moon. During the year, the Sun reaches extreme solstice points and during the lunar month, the phases indicate where the sun is relative to the sun: their configuration relative to one another, leading to stronger or weaker tides.

The tides therefore vary but when the lunar orbit is in phase with the solar ecliptic path, the moon rises above and below that path and the moon becomes more extremely north and south than the solstice sun ever can be. Within a single year, the sun is at winter solstice in midwinter, and summer solstice in midsummer. But the moon takes 18.618 years to reach its maximum standstill, further south and north than the solstice sun.

Ancient cultures were aware of this cycle and sometimes thought to place monuments or burial places on an alignment with maximum moonrise or moonset, occurring north and south of east on the eastern or western horizon.

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Walking on the Moon

There are plans to walk again on the moon (above is a NASA visualization), but there is a sense in which the surface of the moon belongs to the surface of the earth, since the earth’s circumference is 4 times the mean diameter of the earth, minus the moon’s circumference.

The Earth and Moon were formed out of an early collision which left the two bodies in an unusual relationship to one another, in more ways than one. Here we discuss the diameter (and circumference) of each body as a sphere as being in the ratio 11 to 3. The diameter of the Moon is 2160 miles so that the common unit is 720 miles (the harmonic constant) and the diameter of the spherical mean earth would be 7920 miles.

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Working with Prime Numbers

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.

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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.