Author: Richard Heath

Eleven Questions on Sacred Numbers

In 2011, Sacred Number and the Origins of the Universe was nicely re-published in Portuguese by Publisher Pensamento in Brazil. Their press agent contacted my publisher for an email interview from a journalist who posed eleven questions about sacred number.

Interview:

1) Is the universe a mathematical equation?

If the universe is a creation then it needs to have organizing principles governing its structure. I believe that this structure is governed by what we call sacred numbers. Numbers relative to each other form proportions that in sound are perceived as musical intervals. The universe is more like a set of musical possibilities, making it more dramatic and open-ended than an equation.

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Musing Mystical Review: Language of the Angels

I came across a very good review of my book Sacred Geometry: Language of the Angels in which the reviewer found the experience of reading over a month. She also told me how I might improve my handling of numbers. Many thanks for that since I will soon be entering the production phase of a further book out late 23, early 24.

Reading this book has made me grapple with concepts such as the influence of planets on human events, the true nature of the universe, and the magnificent, though forgotten, geometry behind sacred sites. I will admit it took me well over a month to make my way through this book, and oftentimes I had to reread a section multiple times, wondering if I was truly comprehending it. However, it has been a worthwhile pursuit that I’m happy to have made my way through.

Alanna Kali is an astrologer, numerologist, and pioneer spirit that loves to explore life through the lens of depth psychology.
Sacred Geometry, by Richard Heath

Alanna also reviewed the previous book here,

Overall, The Harmonic Origins of the World is an absolutely fascinating book. It is complex, but the daunting amount of information serves as a call to rethink the evolution of consciousness. I recommend this book for anyone looking to expand on their knowledge on how planets affect the world, how math tells its own story of creation, or for anyone who wants to consider the ancient understanding of consciousness viewed through time. The depth of information provided by Heath will be sure to satisfy the curious mind.

SpiralNature.com

Story of Three Similar Triangles

first published on 24 May 2012,

Figure 1 Robin Heath’s original set of three right angled triangles that exploited the 3:2 points to make intermediate hypotenuses so as to achieve numerically accurate time lengths in units of lunar or solar months and lunar orbits.

Interpreting Lochmariaquer in 2012, an early discovery was of a near-Pythagorean triangle with sides 18, 19 and 6. This year (2018) I found that triangle as between the start of the Erdevan Alignments near Carnac. But how did our work on cosmic N:N+1 triangles get started?

Robin Heath’s earliest work, A Key to Stonehenge (1993) placed his Lunation Triangle within a sequence of three right-angled triangles which could easily be constructed using one megalithic yard per lunar month. These would then have been useful in generating some key lengths proportional to the lunar year:  

  • the number of lunar months in the solar year,
  • the number of lunar orbits in the solar year and 
  • the length of the eclipse year in 30-day months. 

all in lunar months. These triangles are to be constructed using the number series 11, 12, 13, 14 so as to form N:N+1 triangles (see figure 1).

n.b. In the 1990s the primary geometry used to explore megalithic astronomy was N:N+1 triangles, where N could be non-integer, since the lunation triangle was just such whilst easily set out using the 12:13:5 Pythagorean triangle and forming the intermediate hypotenuse to the 3 point of the 5 side. In the 11:12 and 13:14 triangles, the short side is not equal to 5.

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On the Harmonic Origins of the World

Does the solar system function as a musical instrument giving rise to intelligent life, civilization and culture on our planet? This 2018 article in New Dawn introduced readers to the lost science of the megalithic – how our ancestors discovered the special ratios and musical harmony in the sky which gave birth to religion and cosmology. The musical harmonies were the subject of my book released that year, called The Harmonic Origins of the World.

After the ice receded, late Stone Age people developed the farming crucial to the development of cities in the Ancient Near East (ANE). On the Atlantic coast of Europe, they also developed a now-unfamiliar science involving horizon astronomy. Megalithic monuments were the tools they used for this, some still seen in the coastal regions of present day Spain, France, Britain and Ireland. Megalithic astronomy was an exact science and this conflicts with our main myth about our science: that ours is the only true science, founded through many historical prerequisites such as arithmetic and writing in the ancient near east (3000- 1200 BC) and theory-based reasoning in Classical Greece (circa 400-250 BC), to name but two. Unbeknownst to us, the first “historical period” in the near east was seeded by the exact sciences of the megalithic, such as the accurate measurement of counted lengths of time, developed by the prehistoric astronomers. With the megalithic methods came knowledge and discoveries, and one discovery was of the harmonic ratios between the planets and the Moon.

The idea that the planets were gods had been born before the ancient world, through the data of megalithic astronomy and this megalithic idea was the basis for the religious ideas of the East. Megalithic astronomy and Near Eastern religious and harmonic ideas have both been written out of our history of civilization, leaving us with enigmatic monuments and ill-defined religious mysteries. How this slighting of our real history happened is perhaps less important than our discovering again the purpose of the megalithic monuments and of those religious ideas that sprang from the discovery that the planets were harmonically related to life on Earth.

Le Menec Alignments indicate a profound astronomical work in the new stone age by 5000-4000 BC. Composite mash up by David Blake using Blender, Google Earth elevation and imagery plus Alexander Thom geometry and digitized stone locations.

Is human history lacking something fundamental?

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Parthenon as a New Model of the Meridian

This was published as The Geodetic And Musicological Significance Of The Shorter Side Length Of The Parthenon As Hekatompedon Or ‘Hundred-Footer’ in Music and Deep Memory: Speculations in ancient mathematics, tuning, and tradition, in memoriam Ernest G. McClain. Edited by Bryan Carr and Richard Dumbrill. pub: Lulu. photo: Steve Swayne  for Wikipedia on Parthenon.

This note responds to Kapraff and McClain’s preceding paper, in which they discover a many-faceted musical symbolism in the Parthenon. Specifically,  Ernst  Berger’s  new measurements include the shorter side of the triple pedestal of the monument as an accurate length to represent one second of the double meridian of the earth. By applying a knowledge of ancient metrology, Anne Bulckens’ doctoral derivations of a root foot can resolve to a pygme of 9/8 feet, of which one second of latitude would contain 90 such feet. However, as a ‘hundred footer’, the foot  length  should  then be 81/80 (1.0125) feet, the ratio  of  the syntonic comma. This would indicate a replacement, by Classical times, of the geographical constant of 1.01376 feet  within the model of the earth since the original model, by the late megalithic, assumed that the meridian was exactly half of the mean circumference of the earth. These alternative geographical constants co-incidentally represent the ubiquitous theme in ancient musicology of the transition between Pythagorean and  Just tunings and their respective commas of Pythagorean 1.01364 … (in metrology 1.01376) and syntonic 81/80 (1.0125).

By Classical times the term hekatompedon or ‘hundred-footer’ had evolved, to describe the ideal dimensionality of Greek peristyle temples. One of the earliest, the Heraion of Samos, came to be 100 feet long by the end of the 8th century[1], in contrast to the surface width of the Parthenon’s stylobyte which had been established as in the range 101.141 (Stuart, c.1750) to 101.341 (Penrose in 1888) feet[2].

Recent measurements in 1982 by Ernst Berger[3] found that the top surface of the stylobyte was just over 101.25 feet wide4 and that the most frequently occurring length was 857.6 mm. Anne Bulckens’[5] corresponding foot measure for this would be a step of 2.5 feet, each of 9/8 (1.125) feet, to within
one part in 2500; a foot length called a pygme within historical metrology, after the size of small men first mentioned when Herakles was travelling back from India6. The shorter ends of the Parthenon’s stylobyte would then be 90 such feet across.

However, should the two ends be divided by 100, the required foot length of 101.25 feet becomes a microvariation of the English foot, namely 81/80 (1.0125) feet, a ratio identical with the syntonic comma. This is another ratio crucial to the history of ancient tuning theory; being found between pure Pythagorean tones (9/8) and their counterparts within just tuning (10/9); when string lengths are given specific whole number lengths to specify their pitches intellectually.

1. Hurwit, Jeffrey M., (1987), The Art and Culture of Early Greece, 1100-480 B.C., Cornell: Ithaca, 74-77
2. Berriman, A.E., (1953) Historical Metrology, London:
Dent. IX, 116-120.
3. Berger, E., ed. (1986) Parthenon-Kongress Basel, 2 Vols, Mainz: Philipp von Zabern.
4. an average noted by Berriman, 119.
5. Bulckens, A.M. (1999) The Parthenon’s Main Design Proportion and its Meaning, [Ph.D. Dissertation], Geelong: Deakin University, 269 pp. ; (2001) The Parthenon’s Symmetry in Symmetry: Art and Science (Fifth Interdisciplinary Symmetry Congress and Exhibition of the ISIS-Symmetry), (Sydney, 2001), no. 1-2, pp. 38-41.
6. Philostrates of Lemnos (c. 190 – c. 230 AD) Imagines Heracles among the Pygmies, see Loeb Classical Library

A recent article by Jay Kapraff and Ernest McClain[7] observes that the width of the Parthenon symbolically defined one second of latitude (taking surface lengths as linear fractions of latitude). This implies the double meridian length was known within 0.003% of its modern estimation.

A geodetic symbolism was apparently given to shorter side length of the Parthenon, making it smaller than it would have been if modelled on the circumference of the earth as one 3,600th of one 360th part of the mean earth. If so, this geodetic meaning of the Parthenon can be compared with monuments built two thousand years earlier, such as Stonehenge and the Great Pyramid of Giza, within which the relationship of the mean earth was specified, relative to the polar radius, using the same metrological system.

The ancient model of the earth, recovered[8] by John Neal[9] and John Michell[10], used three different approximations of π to model the distortion of
the rotating planet relative to its mean, or perfectly spherical, size. In that model, the Meridian was assumed to be half the circumference of the mean earth of 44 times 126 (131,383.296) feet or 24,883.2 miles. Had the Parthenon’s builders used this model then its ends would be 101.376 feet in width and one hundredth of this would be a foot of 1.01376 feet, the foot known as the ‘Standard Geographical’ Greek foot[11].

The mean circumference of the earth (24,883.2 miles) and the actual double meridian length (24,859.868 miles) are in the same ratio as the geographical foot of 1.01376 (3168/3125) and 1.0125 feet: the 81/80 foot measure that makes the Parthenon’s 101.25 feet a ‘hundred footer’. It is therefore reasonable to assume that, between the building of Stonehenge and Great Pyramid (by 2,500 B.C.) and the building of the Parthenon (designed by 447 B.C.), a more accurate
measurement of the Meridian had superseded the previous assumption, within the old model, that the Meridian was half the length of the mean earth circumference.

7. The Proportional System of the Parthenon, in preparation for the In Memoriam volume for Ernest McClain (1918-2014)
8. Michell by 1980 and Neal, fully formed, by 2000.
9. Neal, John (2000) All Done With Mirrors, Secret Academy, London.
10. Michell, John (1982) Ancient Metrology, Pentacle Books, Bristol, 1982; (2008 new ed.) Dimensions of Paradise, Inner Traditions: Rochester.

Further to this, one can see how the transition from Pythagorean to just tuning systems[12] is strangely present in the relationship between the mean earth circumference and the actual meridian length, since the geographical constant of 1.01376 is near identical to the Pythagorean comma of 1.0136433 while the (chosen) ratio of 1.0125 is the syntonic comma and this, times 100, is near identical to the actual length of one second of latitude which would be 100 times 1.0128 feet[13], just one third of an inch different from a more
modern result.

The Parthenon ‘Hundred footer’ was able to dimensionally reference one second of the Meridian by having its shorter sides one hundred feet of 1.0125 feet long. Aligned to north, this presented accurate Classical knowledge of the
Meridian’s length. The monument expresses other musicological features via its metrology: the 81/80 foot unit is 125/128 of the Athenian foot of 1.0368 feet, a musical interval called the minor diesis, also found within just intonation and equaling the deficiency of three major thirds to the octave

12 The latter prevalent in other aspects of the monument, see Kappraff, J. and McClain, E.G. (2005: Spring–Fall) The Proportions of the Parthenon: A work of musically inspired architecture, Music in Art: International Journal for Music Iconography, Vol. 30/1–2.
13 A non-harmonic 79/78 feet.

The Approximation of π on Earth

π is a transcendental ratio existing between a diameter/ radius and circumference of a circle. A circle is an expression of eternity in that the circumference, if travelled upon, repeats eternally. The earths shape would be circular if the planet did not spin. Only the equator is now circular and enlarged, whilst the north and south poles have a shrunken radius and, in pre-history, the shape of the earth’s Meridian between the poles was quantified using approximations of π as was seen in the post before last. In some respects, the Earth is a designed type of planet which has to have a large moon, 3/11 of the earth’s size and a Meridian of such a size that the diverse biosphere can be created within the goldilocks region of the Sun’s radiance.

It would be impossible to quantify the earth as a physical object without the use of approximations to π, a technique seen as emerging in Crucuno between its dolmen and famous {3 4 5} Rectangle where the 32 lunar months in 945 days was used, through manipulation of proximate numbers to rationalize the lunar month to 27 feet (10 Drusian steps) within which days could be counted using one Iberian foot (of 32/35 feet) as described here and in my Sacred Geometry book.

John Michell (1983) saw that different types of foot had longer and shorter versions, different by one 175th part and corresponding to the north-south width of two parallels of latitude: 51-52 degrees, which is the mean earth degree, and 10-11 degrees. The ratio 176/175 is interesting as for its primes.

  1. The harmonic primes {2 3 5} are 16/25 times 11/7.
  2. The 11/7 is half of the pi of 22/7 and the harmonic ratio is the inverse of 25/8.

From this it is clear that these two latitudes are related by the approximation to 1 of a π (22/7) and a reciprocal 1/π (8/25).

But John Neal (2000) saw that some feet also expressed 441/440 which is the ratio between the mean radius of the earth and its polar radius, visually clear in the Great Pyramid. This ratio is also the cancellation of two different πs, namely 63/20 and 7/22 since 7 x 63 = 441 and 20 x 22 = 440. From this emerged an ancient model of the earth that was embodied within the ancient metrology itself. I call this the metrological model rather than the (earlier) geometrical model based upon equal perimeters and the singular π of 22/7.

The metrological model gave a set of regular reference latitudes that accurately defined the geoid of the planet’s meridian by 2,500 BC. One can ask how those developing the model came across the idea of using proximate ratios of π like 176/175 and 441/440, since the system works so well that one may say that the meridian appears to have been designed that way.

The geometric model already defined the mean radius as 3960 miles and so that gives a mean earth meridian of 22 x twelve to the power six. One 180th of this gives a degree length of 364953.6 feet and this is only found at the parallel 51-52 degrees. It is this that defines the megalithic in England, Wales, Scotland and Ireland, an obvious candidate for the metrological survey whose complementary latitude was probably 175/176 of this (362880 feet) in Ethiopia, south of the Great Pyramid. The parallel of the Great Pyramid is 441/440 longer (362704.72) than that of Ethiopia while Athens and Delphi are 440/441 of the mean earth and Stonehenge parallel, that is 364126 feet.

This system was first set by Neal in All Done With Mirrors 2000 as I was writing my first book Matrix of Creation. Are we to think Neal made it up or are we dealing with an exact science that had developed through the megalithic enterprise. And if the Egyptians had an exact science of the earth’s geiod, what are we to make of the fact that the earth appears to follow such a numerically inspired pattern of relationships still true today, in the age of global positioning satellites.

One clue lies in the mind, and how ancient number sciences focus holistically upon the balancing mean. A mean earth that did not spin never existed, since it was only the collision with another planet which created the Moon 3/11 smaller than the Earth. The mean earth radius is these days established as the cube root of the equatorial radius squared times the polar radius. This is less, by 3024/3025, than the geometric model’s mean earth radius of 3960 miles, again maintaining rationality.

It would appear that, in entering the physical and spatial, any planetary design might have been based upon precise rational approximations, about the mean size, of π. To this mystery must be added the musical harmony of the outer planets to the Moon, the Fibonacci harmony of Venus to the Earth itself and the extraordinary numerical relationships of planetary time created by the Sun, Moon and Earth documented by my heavily-diagrammed books and website. From this, more and more can be understood about our prehistory and about its monuments.

Books on Ancient Metrology

  1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
  2. Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
  3. Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
  4. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  5. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  6. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
  7. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  8. —-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology.  The Squeeze Press: 2024.
  9. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.