Tag: Pythagoras

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.

Introduction to my book Harmonic Origins of the World

Over the last seven thousand years, hunter-gathering humans have been transformed into the “modern” norms of citizens (city dwellers) through a series of metamorphoses during which the intellect developed ever-larger descriptions of the world. Past civilizations and even some tribal groups have left wonders in their wake, a result of uncanny skills – mental and physical – which, being hard to repeat today, cannot be considered primitive. Buildings such as Stonehenge and the Great Pyramid of Giza are felt anomalous, because of the mathematics implied by their construction. Our notational mathematics only arose much later and so, a different maths must have preceded ours.

We have also inherited texts from ancient times. Spoken language evolved before there was any writing with which to create texts. Writing developed in three main ways: (1) Pictographic writing evolved into hieroglyphs, like those of Egyptian texts, carved on stone or inked onto papyrus, (2) the Sumerians used cross-hatched lines on clay tablets, to make symbols representing the syllables within speech. Cuneiform allowed the many languages of the ancient Near East to be recorded, since all spoken language is made of syllables, (3) the Phoenicians developed the alphabet, which was perfected in Iron Age Greece through identifying more phonemes, including the vowels. The Greek language enabled individual writers to think new thoughts through writing down their ideas; a new habit that competed with information passed down through the oral tradition. Ironically though, writing down oral stories allowed their survival, as the oral tradition became more-or-less extinct. And surviving oral texts give otherwise missing insights into the intellectual life behind prehistoric monuments.

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An Angelic Geometrical Design

The above diagram contains information with can generally only be grasped by using a geometrical diagram. Its focus is the properties of a right triangle that is 4 times larger than its third and shortest side. The left hand view illustrates what we call Pythagoras’ theorum, namely that

“The squares of the shorter sides add up to the square of the longest side.”

Here this is shown as 144 + 9 = 153 because, if the third side is three lunar months long, then the 4-long base is 12 lunar months, hence the square of 12 is 144″. The longest side is then 153, the diagonal of the four squares rectangle, and the square root of 153 is 12.369 lunar months, the solar year when measured in lunar months.

Before Pythagoras, the Egyptians had a long tradition of geometrical mathematics which fed into their art in which designs can be seen to obey a grid of squares. Their view of Pythagoras’ theorum can therefore be put within a greater world of geometrical transforms using grids.

In the above, one can see this view (called Canevas by Schwaller de Lubicz, The Temple of Man) in which the larger square is seen to fit when angled into a 5-by-5 grid (see right). The extra width and height of the grid enables the smallest square to be seen in this common framework of 25 squares.

The largest square of area 153 is distinguished as an integer, rather than its square root. Thus this is not a Pythagorean triangle with all sides integral, but rather the two smaller sides being integer allows them to be placed within a grid. Somewhat rare though is the arising of an integer on the square, so that Jesus disciples in the gospel of John could comment, in being asked to throw their net on the right side, they then caught 153 fish!

If the diagram was in its least numbers, the 153 would be 9 times smaller as 17 and so the 12.369 would be √9 × √17 instead. And in sacred number science, the interaction of numbers can be seen to be determined by the prime numbers which then make larger numbers such as 153 = 9 × 17. This 17 is known to be a factor of the node cycle of 18.618 solar years, which is 6800 days long and 6800 = 400 × 17.

When two lengths of astronomical time share a larger prime such as 17, it indicates numerical compatibility between two periods, and so the solar year of √153 lunar months (in which the sun moves once around the Ecliptic) has some affinity with the 6800-day period during which its orbital nodes also move once through the Zodiac.

If the larger, yellow square has 6800 days within it, the square root is 20 × √17, whilst the square of the solar year had 153, the square root being 3 × √17.

The new imagined diagram would be 20/3 relative to the above one. Without explaining how this could be, the point is that this cannot be known by the human mind without using sacred geometry which can notate how a higher intelligence might have organised the time environment of Earth according to definite criteria. Further examples can be found in my Book, Sacred Geometry: Language of the Angels. The book is not about sacred geometry as a compendium of traditional knowledge but rather shows how it was that sacred geometry came into the human mind (and architecture) through the initial study of time periods as counted lengths, revealing angelic coincidences.

There is much else to know about the lunation triangle linking the lunar and solar years, discovered about 3 decades ago by my brother Robin Heath.

Twelve: determining Time and Space on the Earth

ABOVE: South rose window in Angers Cathedral of Saint Maurice. Stained glass by Andre Robin created after the fire of 1451. At centre, Christ of the Apocalypse, in glory (Revelation 21:5). At bottom, 12 radial windows showing 12 elders, crowned and playing musical instruments, rejoicing, indicating the remade world (the heavenly Jerusalem). At top, circular ends of 12 radial windows showing the 12 signs of the Zodiac, indicating the incarnation of Christ as a man on earth under the stars. Sequence from left to right has last 2 signs before first, i.e. Aquarius/Water-bearer (grey), Pisces/Fish (grey), Aries/Ram, Taurus/Bull (yellow), Gemini/Twins, Cancer/Crab (red), Leo/Lion (yellow), Virgo/Virgin, Libra/Scales, Scorpio/Scorpion, Sagittarius/Archer, Capricorn/He-goat (blue background)
photo: Chiswick Chap for Wikipedia Foundation.

The Moon was the means by which a 12-fold harmony became established on the Earth. This harmonization occurred through the lengthening of the lunar month until 12 months fitted, in a special way, within the solar year. The excess of the solar year over the lunar year of 12 months became 7/19 lunar months, causing seven extra whole months over 19 years. This 19-year (235 month) Metonic period was well-known to the ancient world, and it leads to the remarkably short cycle for the pattern of similar eclipses, we call the Saros period, which repeat every 18 years (235 minus 12 months = 223 months). And eclipses are highly visible because the disk of the Moon has come to be the same angular size as the disk of the Sun, causing total solar eclipses.

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The Tetraktys as plan of planetary harmony and the four Elements

Figure 1 The elimination of 5 as a factor in the harmonic mountain for 36 lunar years, resolved using matrix units of one tenth of a month and the limit 360 units.

In a previous post 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.

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