## 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.

## 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.

NEXT:

#### CONTENTS

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.

#### CONTENTS

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

## Multiple Squares to form Flattened Circle Megaliths

above: a 28 square grid with double, triple (top), and four-square rectangles (red),
plus (gray again) the triple rectangles within class B

#### Contents

1.     Problems with Thom’s Stone Circle Geometries.

2.     Egyptian Grids of Multiple Squares.

3.     Generating Flattened Circles using a Grid of Squares.

#### ABSTRACT

This paper reviews the geometries proposed by Alexander Thom for a shape called a flattened circle, survivors of these being quite commonly found in the British Isles. Thom’s proposals appear to have been rejected through (a) disbelief that the Neolithic builders of megalithic monuments could have generated such sophistication using only ropes and stakes and (b) through assertions that real structures do not obey the geometry he overlaid upon his surveys.

Continue reading “Multiple Squares to form Flattened Circle Megaliths”

## 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.