Only two type-D stone circles (see figure 3) are known to exist, called Roughtor (in Cornwall) and Seascale (in Cumbria). Seascale is assessed below, for the potential this type of flattened circle had to provide megalithic astronomers with a calendrical observatory. Seascale could also have modelled the harmonic ratios of the visible outer planets relative to the lunar year. Flattened to the north, Seascale now faces Sellafield nuclear reprocessing plant (figure 1).

Stone Age astronomical monuments went through a series of evolutionary phases: in Britain c. 3000 BC, stone circles became widespread until the Late Bronze Age c. 1500 BC. These stone circles manifest aspects of Late Stone Age art (10,000 - 4500 BC) seen in some of its geometrical and symbolic forms, in particular as calendrical day tallies scored on bones. In pre-literate societies, visual art takes on an objective technical function, especially when focussed upon time and the cyclic phenomena observed within time. The precedent for Britain's stone circle culture is that of Brittany, around Carnac in the south, from where Megalithic Ireland, England and Wales probably got their own megalithic culture.

### The first steps taken by Megalithic Astronomy

The stone circles, rectangles, rows, menhirs, dolmens and chambered "tombs" of Carnac seem somewhat “whittled down” in mainland Britain to stone circles and alignments, suggesting those styles of monument found only at Carnac were built to discover the celestial time periods and study how they interact with each other. This activity generated those special numbers considered sacred by later cultures. The lunar month and then solar year would be foundational, as their lengths were resolved to ever greater accuracy using day counting.

Firstly, the lunar month needed to be resolved accurately but at 29.53089 days in length, a process of refinement was required; starting with a 59 day estimate between three full moons (two periods). At Le Manio there is some evidence that this process came to an accurate conclusion with the observation that in 945 days there were 32 lunar months, yielding a rational fraction which their integer arithmetic could handle of 945/32 as the month length, in our notation 29.53125 days. The 32nd stone of Le Manio's Quadrilateral (Heath, Richard, 153) gave knowledge of this fact, since it is 32 months from the start of a 36 month (3 lunar year) day-inch count (Heath, Richard, 105).

Secondly, the solar year length could be resolved through day-inch counting. This was the case at Le Manio Quadrilateral where three solar years were captured as a 1095.75 day-inch count, longer than three lunar years of 1063.1 day-inches by 32.625 day-inches (using 1/8th of a day-inch resolution). As a rational fraction, this is 261/8 inches long or 2.71875 feet: quite possibly therefore, the origin of megalithic yard of 2.72 feet (Heath & Heath, 2011). The megalithic yard was identified by Alexander Thom as a prime megalithic building unit used within the stone circle geometries of Britain [Thom, 1967 & 1978], especially in creating “Pythagorean” triangles within egg shaped (or enlarged perimeter) stone circles.

Thirdly, the longest two sides of a right-angled triangle were used to compare celestial time periods. Embodying the trigonometrical functions such as sine, cosine and tangent, such triangles transform the use of exact measures to hold counted numbers as lengths. The most significant triangle in this respect was intuited as implicit within the second Pythagorean triangle by Robin Heath [1993]. An exemplar monument for this triangle was only identified in Carnac in 2009 within the Le Manio Quadrilateral. Our 2010 survey confirmed its accuracy whilst confirming my own hypothesis at that time, that the megalithic must have used day-inch counting to initially achieve its results.

Further to this, a local group (Fleury et al, 1977) had intuited the widespread use of multiple squares around Carnac (for summary see Heath, 2014, chapter 2] and, in that context, in the search for a triangle with base = 12 and shortest side = 3 units, one can also look for half of a four-square rectangle length 12 lunar months and side 3 lunar months. The four-square geometry is an easier framework to construct and building it gives a very high accuracy in reproducing the solar year to lunar year ratio. Four squares would have rendered the lunation triangle portable to other regional contexts.

### The evolution of stone circle observatories

In this article I will show that the type-D flattened circle, as defined by Thom, could have evolved from the four-square device discovered at Le Manio to provide such a circle’s builders with the calendrical geometry of the Sun and Moon. Whilst Le Manio was probably c. 4000 BC +/- 200 years, Stonehenge 1 is dated around c. 3000 BC +/- 100 years and so, what difference did 1000 years make to the context in which the 4-square rectangle was used?

There is clear evidence of Thom's egg-shaped circles at Carnac and hence stone circle design was present there in c. 4000 BC in Carnac before becoming a norm for Scotland and England c. 3000 BC.

It is interesting to look at the apparently new, flattened designs, not found in Carnac. Thom estimates: "The majority of the rings [known anywhere] were true circles but in Fig. 3.1 we give for reference the commoner variations. Of the 900 Megalithic rings known to Burl he thinks that about 600 are circles, 150 are flattened circles, 100 are ellipses and 50 are egg-shaped (Burl, 1976). There were also the compound rings (Thom 1967) leading finally to the most advanced of all, that at Avebury." [Thom 1978. 17]

### Type-B stone circles as calendrical devices

Flattened circles (i, ii, iii and iv above) are symmetrical along one axis (shown vertical in figure 3) and have a diameter (M-N) longer than the V-shaped long radius (A-B) that stands in an invariant ratio (A-B/M-N). The long radius of the type-B, easiest to construct using ropes, immediately relates to the counting of time since its diagonal is a 3-square rectangle (Thom, 28-29), whilst also a vesica pisces geometry [Robin Heath, 1998.102-3].

Less astronomically accurate than a 4-square rectangle, the 3-square rectangle relates two pairs of celestial time periods about the solar year. If the longest side (or hypotenuse) is the solar year (of 365.242 days) in day-inches then the base side equals the eclipse year (of 346.62 days) in day-inches. If instead, the solar year is the length of the base then the hypotenuse is the length of the 13-month lunar year (of 383.898 days) in day-inches. These two triangles can therefore be "stacked up" since the solar year is common to both relationships as shown in figure 4 below.

A type-B flattened circle, once constructed, is evidently capable of hosting a framework for day counting: The ratio of solar year / eclipse year is 1.053725 (and this forms an acute triangular angle of 18.375 degrees) whilst the angle of the diagonal of a 3-square forms a 18.435 degree angle, so that a perfect solar year gives an eclipse year 3 hours short of 346.62 days and a 13-month year that is a full day longer. The 3-square is therefore a better calendrical device between the eclipse and solar years than between the solar year and 13-lunar month year.

Whatever its other uses might have been, the type-B geometry could clearly provide a calendrical function for the day-counting astronomers of the megalithic, interested in eclipses.

If the long V-shaped radius of the type-B was the solar year, then the two extreme points, where that radius forms an arc, are the corners of two vertical 3-square triangles which end at point A (figure 5).

Thom studied the possible meaning of such flattening. For example, the new perimeter relative to the original diameter M-N gives a kind of pi equal to 2.9572, and Thom thought this a possible attempt to approximate pi as 3 [Thom, 1967: 28-29. 1978, 17]. Another use for the type-B was proposed by Robin Heath [1998. 102] where he assumed the original circle was the solar year (e.g. 365.2422 day-inches in circumference), in which case the flattened perimeter would be 343.666 days, short of 346.62 day-inches by 125/126**.

**[a known microvariation applied to all the feet made available by ancient metrology [Neal, 2000]. If MN was 116.213 standard canonical Greek inches (126/125 larger than the inch), then the perimeter of the type-B would be 346.415 day-inches, of the ordinary kind.]

The precedent for embedding a calendrical device within a flattened circle is therefore found in the reasonably widespread type-B circles. When interpreting their metrology, one should seek to find units of measure which might have enabled them to be used as a calendric device. And the question naturally arises; did any of Thom's other flattened designs embed calendric devices?

### Type-D stone circles as calendrical devices

I recently interpreted the Parthenon as manifesting ancient-world knowledge of how the outer planets form harmonic ratios to the lunar year [Heath 2018. 72-78]. I also suggested this knowledge came from the astronomers of Atlantic Europe, perhaps explaining the mythic purpose of Plato's Atlantis story. Until recently I was unable to find substantiating evidence for such a harmonism within megalithic monuments though I have provided a few possible interpretations online [Heath, Richard, 174]

To find an example as compelling as the Parthenon, a monument needs to provide at least two counted lengths including the lunar year and at least one synod of Jupiter or Saturn, as separate lengths. I recently found such in Thom’s type-D flattened circle specifically at Seascale, where the forming circle is 90 Roman feet (of 0.96768 ft) in diameter (Thom's M-N). In Thom’s geometry, the circle can *almost* accommodate a 4-square rectangle in its V-shaped component, and were it to, A-B would be the solar year (the diagonal of the 4-square, and hypotenuse of Robin Heath's lunation triangle).

The long side of the 4-square is then the lunar year, which would be 80 Roman feet in length, the number of matrix units (see later) that divides up the lunar month in order to make the lunar month commensurate with the synods of the outer planets. Jupiter's synodic period of 398.88 days is accurately 9/8 of the lunar year, a relationship directly available between the 90 foot diameter of the Seascale stone circle and the implicit 4-square rectangle of the type-D design if and when designed with a diameter of 90 feet so that an exact 4-square rectangle of 80 feet could be vertically mounted within it**.

**[The megalithic astronomers did not have to follow any guidance but their own to creatively realising the stone circles or other "monuments". The tendency to either rubbish Thom's geometries or to slavishly insist upon them ignores the fact that Thom’s design rules were speculative yet successful in providing a taxonomy for stone circles.]

There were three significant dimensions:

1. The diameter of the circle M-N of 90 Roman feet representing the Jupiter synod,

2. The height of the 4-square rectangle of 80 Roman feet, representing the lunar year and

3. The V-shaped radius from bottom centre A-B (diagonal of the 4-square) of sqrt(6800) =82.46 Roman feet, representing the solar year.

The metrology is M-N = 90 and A-X = 80. At the centre is a 40 by 80 rectangle running from A in which two 4-squares provide diagonals that give the V-shaped radius A-B.

Before moving on into the harmonic significance of the Seascale arrangement, attention should be given to the lunation triangle at its heart. Such triangles, of differing ratios, can in general be normalised through dividing their sides by their difference in length, a difference which, in the case of three solar and lunar years, equals the length of the megalithic yard (32.625 day-inches) - as was seen at Le Manio's Quadrilateral (Heath & Heath, 2011). The lunation triangle was probably the foundational tool for megalithic investigations which used day-counted triangles to compare celestial periods.

In the British megalithic, the megalithic yard was used to count months instead of days so as to reduce the yearly differential to the length of our English foot [Robin Heath, 1998. 82] since that yard in feet (2.72) is the reciprocal of a single solar year’s excess, in lunar months (0.368), over the lunar year. This meant that when the times between the synodic loops of the outer planets was counted in months, these were immediately 12.8 months for Saturn and 13.5 months for Jupiter. Relative to the lunar year of 12 months, Saturn was 16/15 of the lunar year whilst Jupiter was 9/8 of the lunar year.

### The Harmonic Framework of the Outer Planets

The symmetrical axis of a flattened circle can be seen to be a perfect place to vary the length of A-B, through forming additional triangles that compare the synods of the outer planets with the solar and lunar year, already present in a lunation triangle geometry.

At Seascale (figure 9), the Jupiter synod can be built into ratio between the forming circle diameter M-N and the height of the 4-square rectangle. The Jupiter synod relative to the solar year is also visible between M-N, as a vertical diameter, and A-B, the length of the solar year. An arc from point A of length M-N would form a right-angled triangle if the top of the 4-square is extended to the point where the arc crosses it. The ratio of side lengths is then 90/80 or 9/8, the Pythagorean whole tone. The units are 1.5 lunar months since 8 x 1.5 lunar months equals the 12 lunar months of the lunar year (here 80 Roman feet). The type-D flattened circle can then be seen (as with the type-B) as a stone circle framework for considering multiple time periods along its axis of symmetry.

Whilst a foot now called Roman was employed at Seascale, this was long before the Roman empire existed. Such “historical” measures were named according to the region in which they were first found by the historical metrologists, measures being found within measuring sticks, sculped body parts, or "itinerary" lengths such as furlongs or miles of a given foot. The Roman foot used at Seascale was first found by Greaves within the statue of Cossutius: it is called pes Colitinus and equals 0.967 feet. Below is a Roman foot rule as one might pick up in a shop near Hadian's Wall.

The rule (figure 7) tells us the Roman foot, like the English, was made up of 12 inch divisions so that; if the 4-square at Seascale is 80 Roman feet long then it is 960 inches long. 960 is a key number in resolving the harmonic ratios between the outer planets and the lunar year by looking at all the harmonic numbers less than 1440**, the same limit used at the Parthenon to express the same knowledge.

**[implicit in the name of Adam [ since A.D.M = 1 + 4 + 40 = 45 and 1440] within a harmonically encoded Bible whose writers must have had knowledge of the synodic harmonism between the Moon and outer planetary synods.[Heath, 2018.]]

1440 is the smallest numerical context for the lunar year to be 960, Saturn to be 1024, Jupiter to be 1080 (and Uranus to be 1000), when resolved within the ancient technique of harmonic numbers [Heath, 2018], in the manner of ancient harmonists (see figure 8). If 1024 is divided by 12, then A-B can be extended to 1024/12 = 85 and 1/3rd Roman feet to show its 15:16 semitone relationship to the lunar year (figure 9).

Whilst Uranus (=1000) is difficult to discover using naked eye astronomy, Uranus’s synod is implicit in the 80 units of the 4-square at Seascale in that 1000 such units of 29.53125/ 80 = 0.36914 days times 1000 equals 369.14 days, half a day less than the Uranus synod of 369.66 days. The third outer planet is therefore implicit in the matrix unit of 0.36914 days.

Somewhere in the near east ancient harmonic thought appears to have re-presented the harmonic relationships of outer planets and lunar year, arithmetically rather than geometrically. We can provide the holy mountain** (figure 8) for the limiting number 1440 (= 18 lunar months), which shows these and other relationships as part of the type of tuning environment called Just intonation, within which the ancient world generally tuned their instruments [McClain, 1978].

**[implicit in the name of Adam [ since A.D.M = 1 + 4 + 40 = 45 and 1440] within a harmonically encoded Bible whose writers must have had knowledge of the synodic harmonism between the Moon and outer planetary synods [Heath, 2018.].]

### In conclusion

We saw earlier that type-B flattened circles were natural calendrical devices, linking the solar and eclipse years using a three-square geometry, when laid-out as Alexander Thom envisaged they were, with ropes and pegs. It also appears that a calendric device could be constructed in which the four-square geometry could define the flattening of the type-D circle at Seascale and Roughtor, and this would then only slightly deviate from Thom’s design. We have also seen that both these types of circle were ideal (in their flattening) as a common and elegant framework for the comparison of more than two time periods, using right-angled triangles with a common origin at A.

This is true of both type-A and type-B designs, exactly because the centre of the smaller radius arc (which provides continuity to their flattened perimeter) is located where the radiants from the circle’s centre and the radiants from point A, cross. This avoids any need to define this point geometrically, as Thom did, because the V-shaped radiants will *inevitably* cross either M-N (type-B) or whatever radius terminates the circular portion of their design (type-A). So, whilst Thom saw the problem as requiring a fixed solution, *any radii from A*, however constructed, will cross any radii from the centre. This means that one can place any triangle, whose hypotenuse is in a wide range of ratios to its base, and generate a flattened stone circle but, the 3 and 4-square designs give counting of the sun and moon cycles so that the outer planetary synods can then also be projected, from point A, into the flattened region.

To illustrate this, figure 10 turns a diagram I made years ago by 90 degrees and lays it upon the survey for Seascale.

Flattened circles could simply have been ways of combining the familiar functions of a stone circle, such as symbolic alignments to horizon events or the creation of a 12-fold structure to measure angles, with an observatory’s need to maintain a calendar of relevant celestial cycles. I have also suggested how, facing north, flattened circles could be used to observe the circumpolar stars to measure time within the night [Richard Heath, 106].

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