Counting the Moon: 32 in 945 days

One could ask “if I make a times table of 29.53059 days, what numbers of lunar months give a nearly whole number of days?”. In practice, the near anniversary of 37 lunar months and three solar years contains the number 32 which gives 945 days on a metrological photo study I made of Le Manio’s southern curb (kerb in UK) stones, where 32 lunar months in day-inches could be seen to be 944.97888 inches from the center of the sun gate. This finding would have allowed the lunar month to be approximated to high accuracy in the megalithic of 4000 BC as being 945/32 = 29.53125 days.

Silhouette of day-inch photo survey after 2010 Spring Equinox Quantification of the Quadrilateral.

One can see above that the stone numbered 32 from the Sun Gate is exactly 32/36 of the three lunar years of day-inch counting found indexed in the southern curb to the east (point X). The flat top of stone 36 hosts the end of 36 lunar months (point Q) while the end of stone 37 locates the end of three solar years (point Q’). If that point is the end of a rope fixed at point P, then arcing that point Q’ to the north will strike the dressed edge of point R, thus forming Robin Heath’s proposed Lunation Triangle within the quadrilateral as,

points P – Q – R !

In this way, the numerical signage of the Southern Curb matches the use of day-inch counting over three years while providing the geometrical form of the lunation triangle which is itself half of the simpler geometry of a 4 by 1 rectangle.

The key additional result shows that 32 lunar months were found to be, by the builders (and then myself), equal to 945 days (try searching this site for 945 and 32 to find more about this key discovery). Many important numerical results flow from this.

Cologne Cathedral Facade as Double Square

image: The Gothic cathedral of Cologne by night, by Robert Breuer CC-SA 3.0

On the matter of facades of Gothic cathedrals, I hark back to previous work (February 2018) on Cologne cathedral. This was published in a past website that was destroyed by its RAID backup system!

As we have seen with Chartres, some excellent lithographs with scales can often exist online from which one can interpret their sacred geometrical form and even the possible measures used to build that form. The Gothic norm for a facade seem more closely followed at Cologne facade which has two towers of (nearly) equal height.

We saw at Chartres that an underlying geometry using multiple squares may have been used to define a facade and bend it towards a suitable presentation of astronomical time, in a hidden world view that God’s heaven for the Earth is actually to be found in the sky as a pattern of time. This knowledge emerged with the megaliths and, in the medieval, it appeared again in monumental religious buildings built by masons who had inherited a passed-down but secret tradition.

A Prologue to Cathedral Music

Continue reading “Cologne Cathedral Facade as Double Square”

Using Circumpolar Marker Stars

The marker stars within the circumpolar or arctic region of the sky have always included Ursa Major and Ursa Minor, the Great and Little Bear (arctic meaning “of the bears” in Greek), even though the location of the celestial North Pole circles systematically through the ages around the pole of the solar system, the ecliptic pole. In 4000 BC our pole star in Ursa Minor, called Polaris, was far away from the north pole and it reached a quite extreme azimuth to east and west each day, corresponding to the position of the sun (on the horizon in 4000 BCE at this latitude) at the midsummer solstice sunrise. This means angular alignments may be present to other important circumpolar stars in some of the stones initiating the Alignments at Le Menec, when these are viewed from the centre of the cromlech’s circle implicit in its egg-shaped perimeter.

This original “forming circle” of the cromlech could be used as an observatory circle, able to record angular alignments. Therefore the distinctive “table” stone which aligns to the cromlech’s centre at summer solstice sunrise, also marked the extreme angle (to the east) of Polaris, alpha Ursa Minor, our present northern polestar. That is, in 4000 BCE Polaris stood directly above the table stone, once per day – whether visible or not.

Such a maximum elongation of a circumpolar star is the extreme easterly or westerly movement of the star, during its anti-clockwise orbit around the north pole. Thus, if the northern horizon were raised (figure 5) until it passed through the north pole, the maximum circumpolar positions for a star to east and west would be equally spaced, either side of the north pole. If these extreme positions are brought down to the Horizon in azimuth, the angles between these extremes forms a unique range of azimuths on the ground between (a) the horizon (b) a foresight such as a menhir and (c) an observer at a backsight. Observations of these extreme elongations naturally enable the pole (true north) to be accurately established from the observing point as the point in the middle of that range. A marker stone can usefully locate a circumpolar star at one of these maximum elongations and come to symbolize that important star. A star’s location could have been brought down to the horizon using a vertical pole or plumb bob, between the elongated star and the horizon, at which point menhirs could later be placed, relative to a fixed viewing centre or backsight. This method of maximum elongations would have escaped the atmospheric effects associated with observing stars on the horizon which causes a variable angle of their visual extinction below which stars disappear before reaching the horizon.

Figure 5.The Maximum Elongation of Circumpolar Stars is a twice daily event when, looking at the horizon, the star’s circumpolar “orbit” momentarily stops moving east or west at maximum elongation in azimuth and reverses its motion.

At Le Menec the azimuths of the brightest circumpolar stars, at maximum elongation, appear to have been strongly associated with the leading stones of the western alignments (see figure 6). However, it is likely that only one of these circumpolar stars was used as a primary reference marker, for the purpose of measuring sidereal time at night when this star was visible.

Figure 6 Some of the associations between circumpolar stars and stones in the western alignments. These alignments are all to the maximum easterly elongations, perhaps established during the building of the sidereal observatory and only later formalized into leading stones at the start of different rows. Dubhe was then selected as the primary marker star for the Le Menec observatory.

To achieve continuous measurements of sidereal time from the circumpolar stars requires a simple geometrical arrangement that can draw down to earth the observed position of maximum elongation to east and west for one bright circumpolar star, the observatory’s marker star. A rectangle must then be constructed to the north of the cromlech’s east-west diameter and containing within it the observatory’s northern semicircle. The northern corners must align with, relative to the centre of the circle, the eastern and western elongations of the chosen marker star. For Le Menec the rectangle had to be extended northwards until it reached the first stone of row 6[1]. This stone is aligned, from the centre, to the maximum eastern elongation of Dubhe or alpha Ursa Major. The first stone of row 6 is therefore the menhir marking Dubhe. To the south, the initial stones of further rows all stand on the eastern edge of this rectangle, so that any point on the rectangle’s north face could be brought down, unobstructed, to the circumference of the circle.

Figure 7 shows how the form of the circumpolar region, within the “orbit” of Dubhe, is repeated by the cromlech’s forming circle. It is also true that the “northern line” then has the same length as the diameter of the forming circle, which has therefore been metrologically harmonized with row 6’s initial stone and the alignment to Dubhe in the east.

This arrangement has the consequence that wherever Dubhe is (above the northern line and when seen on a sightline passing through the centre of the cromlech) its east-west location in the sky can be brought down, directly south, to two points on the forming circle of the observatory – all due to the star observation having been made upon a length equal to the circle’s diameter (the Northern Line of figures 7 and 8). One of these two points, on the northern or southern semicircle of the observatory, must then correspond exactly to where Dubhe is in its “orbit” around the north pole, as in figure 8.

So, what is being measured here and what would be the significance of having such a capability? Whilst the movement of all the stars is being accurately measured, using this northern line and forming circle combination, the monument also has a reciprocal meaning. The forming circle also represents the earth’s rotation towards the east, the cause ofthe star’s apparent motion. This is because, when looking north, the familiar direction of rotation of the stars, when looking south, is reversed from a rightwards motion to a leftwards, anticlockwise motion. Circumpolar motion therefore directly represents the rotation of the earth. The Dubhe marker star would have represented the movement of a point on the surface of the earth, moving forever to the east. Perhaps more to the point, the eastern and western horizon are moving as two opposed points on its circular path, each moving at about the same angular speed as Dubhe. This deepens the view of the forming circle as representing those ecliptic longitudes in which the fixed stars, rising or setting on the eastern and western horizons, are fixed locations on the circle through which these horizons are moving as markers on the circle’s circumference.

These two views, of a moving earth and of a moving background of stars, could be interchangeable when understood and both viewpoints are equally useful and were probably relevant to the operation of this observatory. Whilst the circumpolar stars move around the pole, the eastern and western horizon move opposite each other, running along the ecliptic, as the Earth rotates. The first view enables an act of measurement which would have given astronomers access to sidereal time and the second view provided knowledge of where the eastern and western horizons were located viz a vis the equatorial stars and therefore knowledge of which part of the ecliptic was currently rising or setting.

Figure 8 Recreating the circumpolar region with marker star Dubhe at the correct angle on the forming circle of the western cromlech. The star’s alignment on the northern line is dropped to the south so as to touch the two points of the circumference corresponding to that location on the circle’s diameter: one of these will be the angle of Dubhe as seen within the circumpolar sky but now accurately locatable in angle, on the observatory circle.

Dubhe had, in 4000BCE, a fortunate relationship to the circumpolar sky and equatorial constellations which would have been very useful. When Dubhe reached its maximum eastern elongation (marked by the first stone in the sixth row) the ecliptic’s summer solstice point was rising in the east. However, Dubhe’s maximum western elongation did not correspond to the winter solstice, this due to the obliquity of the ecliptic relative to north. It is the Autumn Equinoctal point of the ecliptic that is rising to the east at Dubhe’s maximum western elongation. It was when Dubhe was closest to the northern horizon, that the other, winter solstice point was found rising on the ecliptic. It is important to realize that these observational facts were true every day, even when the sun was not at one of these points within the ecliptic’s year circle.



This paper proposes that an unfamiliar type of circumpolar astronomy was practiced by the time Le Menec was built, around 4000 BCE.

  1. Abstract
  2. Start of Carnac’s Alignments
  3. as Sidereal Observatory
  4. using Circumpolar Marker Stars
  5. dividing the Circumpolar stars
  6. maintaining Sidereal Time in Daylight
  7. measuring the Moon’s Progress
  8. as Type 1 Egg
  9. transition from Le Manio
  10. the Octon of 4 Eclipse Years
  11. building of Western Alignments
  12. key lengths of Time on Earth

[1] Thom’s row VI.

Le Menec: as Sidereal Observatory

Today, an astronomer resorts to the calculation of where sun, moon or star should be according to equations of motion developed over the last four centuries. The time used in these equations requires a clock from which the object’s location within the celestial sphere is calculated. Such locations are part of an implicit sky map made using equatorial coordinates that mirror the lines of longitude and latitude. Our modern sky maps tell us what is above every part of the earth’s sphere when the primary north-south meridian (at Greenwich) passes beneath the point of spring equinox on the ecliptic. Neither a clock, a calculation nor a skymap was available to the megalithic astronomer and, because of this, it has been presumed that prehistoric astronomy was restricted to what could be gleaned from horizon observations of the sun, moon, and planets.

Even though megalithic people could not use a clock nor make our type of calculations, they could use the movement of the stars themselves, including the sun by day, to track sidereal (or stellar) time provided they could bring this stellar time down to the earth. This they appear to have done at Le Menec, using the cromlech’s defining circle, which was built into its design so as to become a natural sidereal clock synchronized to the circumpolar stars.

Figure 4 The Circumpolar Stars looking North from Le Menec in 4000 BCE, when the cromlech was probably built. There is no north star but marker stars travel anti-clockwise and these can align to foresights at their extreme azimuthal “elongation”, as explained below.

The word sidereal means relating to stars and, more usually, to their rotation around the earth observer as if these stars were fixed to a rotating celestial sphere. This rotation is completely reliable as a measure of time since it is stabilized by the great mass of the spinning earth. However, in a modern observatory this sidereal time must be measured indirectly using an accurate mechanical or electronic clock. These clocks can only parallel the rotation of the earth in a sidereal day, which is just under four minutes less than our normal day. Nonetheless, a sidereal day is again given 24 ‘hours’ in our sky maps and it is these hours which are then projected upon the celestial sphere as hours (minutes and seconds) of Right Ascension, hours in the rotation of the earth during one sidereal day.

NEXT: using Circumpolar Marker Stars


This paper proposes that an unfamiliar type of circumpolar astronomy was practiced by the time Le Menec was built, around 4000 BCE.

  1. Abstract
  2. Start of Carnac’s Alignments
  3. as Sidereal Observatory
  4. using Circumpolar Marker Stars
  5. dividing the Circumpolar stars
  6. maintaining Sidereal Time in Daylight
  7. measuring the Moon’s Progress
  8. as Type 1 Egg
  9. transition from Le Manio
  10. the Octon of 4 Eclipse Years
  11. building of Western Alignments
  12. key lengths of Time on Earth

Dun Torcuill: The Broch that Modelled the World

image above courtesy Marc Calhoun


This video introduces an article on a Scottish iron-age stone tower or brock which encoded the size of the Earth. 

You can view the full article on sacred dot number sciences dot org, searching for BROCK, spelt B R O C H.

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 as proposed by Mackie).

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, the simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods., 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.

If we take the size of the moon in that model, as being 3/11 of 396 feet this would give a circle radius 108 feet and one can see that, using the moon, the outer perimeter of the brock was probably elliptical too.

Thank you for watching.

OpenAI: Problems for Research using ChatGPT

I asked the chatbot a series of questions which homed in on a key problem with the chatbot. The problem was that the chatbot thinks that an 819-day period can equal the 260-day period of the Maya calendar period of 260 days.

how does the 819-day period interact with the 260-day Tzolkin?

The 819-day period is the length of a full cycle of the Tzolkin, which is a 260-day sacred and divinatory cycle in the Maya calendar system.

The Tzolkin is formed by the combination of two smaller cycles: a 13-day cycle and a 20-day cycle. Each day in the Tzolkin is represented by a unique pair of glyphs, resulting in a total of 260 possible combinations. The Tzolkin was used for divination, determining the best times for planting crops, making decisions, and performing rituals.

The interaction between the 819-day period and the 260-day Tzolkin is that every 819 days, the Tzolkin cycle completes a full cycle of 260 days, which means that the same combination of the 13-day and 20-day cycles repeats. In other words, every 819 days, the Maya believed that the energies and personalities associated with each of the 260 possible combinations of glyphs would repeat.

This repetition was seen as a symbol of the cyclical nature of time and the universe by the ancient Maya. The Tzolkin was also used to calculate longer periods of time, such as the Haab’ (365-day) calendar and the Calendar Round (a 52-year cycle), which were used for agricultural and religious purposes.

ChatCPT is an artificial intelligence (AI) system which is language based, so that it apparently does not apparently check any mathematical consistency, and this makes it possible for such an error impossible to humans who know that one number never equals another, by definition.

In some areas this AI appears superb, for example in summarizing an issue such as

What were the deficiencies of the Short Stirling Bomber?

The Short Stirling was a British four-engined heavy bomber aircraft used by the Royal Air Force (RAF) during World War II. While it was an important aircraft in the early years of the war, it was ultimately replaced by more advanced designs. The Stirling had several deficiencies that limited its effectiveness, including:

  1. Range: The Stirling had a limited range, which made it difficult to reach targets deep in Germany. This was due in part to its large size and weight, which made it less efficient than other bombers in terms of fuel consumption.
  2. Payload: Despite its large size, the Stirling had a relatively small bomb load compared to other heavy bombers of the time. This limited its effectiveness as a strategic bomber.
  3. Maneuverability: The Stirling was not very maneuverable, which made it vulnerable to enemy fighters. This was partly due to its size and weight, which made it slower and less agile than other aircraft.
  4. Maintenance: The Stirling was difficult to maintain, which meant that it spent more time on the ground than in the air. This limited its operational effectiveness and made it more vulnerable to attack while on the ground.
  5. Visibility: The Stirling had poor visibility from the cockpit, which made it difficult for the crew to spot enemy fighters and defend the aircraft.

Overall, the Stirling was an important aircraft for the RAF during the early years of World War II, but its deficiencies limited its effectiveness as the war progressed. It was ultimately replaced by more advanced designs, such as the Avro Lancaster and the Boeing B-17 Flying Fortress.

Of course it may have found one or more perfect answers to this question.


But in another area, more serious errors came for ancestry research where it can confuse a name and date with another well-formed biography for inexplicable reasons. My wife has a relative Charles Samuel Weston, a family description of whom exists. He was significant as a soldier in colonial India but asking of him first gave:

Charles Samuel Weston (1821-1880) was a British engineer who made important contributions to the development of the telegraph and submarine cables.

Weston was born in Bristol, England and worked as an apprentice at a local brass works before starting his own engineering business. He became interested in the telegraph and developed a number of innovations that improved the reliability and efficiency of telegraph systems.

One of Weston’s most important contributions was the development of a new type of battery, known as the “gravity cell,” which was used to power telegraph systems. The gravity cell was more efficient and reliable than other batteries of the time, and it became widely used in telegraphy and other applications.

Weston also played a key role in the development of submarine telegraph cables, which were used to transmit messages across the Atlantic Ocean and other bodies of water. He developed a number of innovations that helped to make submarine cables more reliable and efficient, including a new type of cable insulation and a system for detecting breaks in the cable.

In addition to his work in engineering, Weston was also a noted philanthropist who supported a number of charitable causes. He died in 1880 at the age of 59.

In changing the date of death to 1909, which is correct,

I apologize for my previous response – there appears to be some confusion regarding the dates of Charles Samuel Weston.

Charles Samuel Weston (1821-1909) was a British engineer and inventor who made significant contributions to the development of the electric telegraph and the submarine cable.

Born in Bristol, England, Weston began his career working as an apprentice at a brass foundry before starting his own business as a mechanical engineer. He became interested in the telegraph and made several important improvements to telegraph systems, including developing a more reliable and efficient battery known as the Weston cell.

etc, then Weston died in London in 1909 at the age of 88.

Before this he had been an administrator who introduced taxation to India, established university, and so on. Compelling but false, and so showing an excellent ability to invent plausible biographies based upon fragments of existing biographical texts.

Charles Samuel Weston was a British civil servant who served in India during the late 19th and early 20th centuries. He played an important role in the administration of the British Raj, particularly in the areas of finance and taxation.

Weston began his career in India in the mid-1880s, working in the Madras Presidency as a deputy collector of customs and salt. He quickly rose through the ranks, serving in various administrative and financial positions in Madras and other parts of the country.

One of Weston’s most significant contributions to British rule in India was his work on the Indian Income Tax Act of 1886. This legislation established a system of income taxation in India, which helped to provide a stable source of revenue for the British government and modernize the Indian economy.

In addition to his work on taxation, Weston was also involved in various other aspects of British administration in India. He served as a member of the Viceroy’s Legislative Council and as a member of the Indian Finance Committee. He was also involved in the establishment of several government institutions in India, including the Imperial Bank of India and the Indian Institute of Science.

Overall, Charles Samuel Weston played an important role in the British Raj in India, helping to modernize the Indian economy and establish systems of governance and taxation that would shape the country for years to come.

Who is Richard Heath?

This question managed to find data despite having no Wikipedia page.

Richard Heath is a writer, researcher, and speaker who is known for his work on ancient cultures, mythology, and symbolism. He has written several books, including “Sacred Number and the Origins of Civilization”, “The Harmonic Origins of the World”, and “Matrix of Creation: Sacred Geometry in the Realm of the Planets”.

Heath’s work explores the connections between ancient sacred sites, mythology, and the principles of sacred geometry and numerology. He has studied the geometry and symbolism of numerous ancient sites around the world, including Stonehenge, the Great Pyramid of Giza, and the temples of ancient Egypt.

Heath’s work has been praised for its interdisciplinary approach, combining insights from archaeology, anthropology, mythology, and geometry. His research has also been influential in the fields of alternative history and alternative archaeology.

In addition to his writing and research, Heath has also given lectures and workshops on topics related to sacred geometry, ancient wisdom, and symbolism.

I need to say that this was remarkable unless it knows I am asking, since there are many Richard Heaths both living and dead including a congressman, Victorian author, furniture shop and so on. One can say this is a novel type of searching which is more interesting and humanistic despite being detached from some kinds of definitional rigour found in non-language artificial intelligence, for instance that 260-days cannot equal 819-days. Many Maya textbooks are not available online so that exotic periodicities, in books like

Star Gods of the Maya: Astronomy in Art, Folklore and Calendars (The Linda Schele Series in Maya and Pre-Columbian Studies) by Susan Milbrath are somewhat hidden to AI while articles such as Scientists Think They’ve Finally Figured Out How a Maya Calendar Works will feature the 819-day period, and other articles will lie behind paywalls. This latter article is not as pedagogical as I have found it necessary to be regarding numbers in ancient calendrical contexts. That is, on a Monday morning, I find it likely that the logic is itself obscure to a general audience. I will endeavour to post on this article this week.