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One of the key objections for the megalithic building flattened circles concerned their use of ropes and a knowledge of geometry in managing radii to achieve a lesser circumference than a circle would have. If instead Thom's Type A or Type B flattened circles were constructed using a grid of squares, then some of the all-important key points where a flattened circle's radius of curvature changes (of which there are only four) should be points of intersection within the grid intersections. It became clear that this was a possible alternative means to their production when considering the Type A geometry and specifically its implicit pair of triple-square triangles, as right triangles available within such a grid.
Robin Heath has already noted [in Sun Moon and Earth, p52-55] that these triangles are close to the invariant ratio, in their longest sides, of
- the eclipse year and solar year, and
- this same ratio is also to be found between the solar year and the thirteen lunar month year.
The baseline of such a right triangle is found to be 6/7 of the diameter MN of the Type A flattened circle and this implies, given the left-right symmetry of this form, that the key point at the end of the hypotenuse (where the radius of curvature changes) would sit on the corner of a grid point of 14 by 14 squares, as a length then equal to twelve grid units. The forming circle used by Thom, of diameter MN, would then inscribe the grid square.
Figure 1 Type A drawn on a 14 square grid
We also know, from Carnac, that the astronomers used a triple square to frame this right triangle so as to relate the periods of eclipse and solar year. Since the vertical position of the key point is 12 units, then to left and right the key points either end of the central flattened arc are 4 units, either side of the central axis. Therefore, to right and left of these triple squares can be found two four-square rectangles, whose diagonals express (with an accuracy better than a day count could could) the relationship of the lunar year (side length = 4) to the solar year (as hypotenuse/diagonal). These four squares (each 3 by 3 = 9 grid squares) have a baseline of twelve grid squares which exactly matches the number of lunar months within the lunar year.
One therefore sees useful megalithic "resources" within such a 14-square grid in that many multiple squares can be formed; such as these triple squares either side of the vertical centreline have two four-square rectangles to the right and left (shown in red below, the ripple-squares being blue). These leave a row of 14 by 2 squares at the top which can be seen as a seven-square, the rectangle whose diagonal to side alignment is found between a double and a triple square: These include triple and four square rectangles which give good approximations in their ratios, between diagonals and longer side lengths, which can be used as calendric devices for lunar to solar year, eclipse to solar year and solar to 13 month year.***
***This habit, around megalithic Carnac, of "finding" right triangles within multiple squares corresponds to the astronomical reality whilst enabling accurate generation of these counted lengths without any day counting of periods; once the triple square and four square were discovered to be "cosmograms"
The other two points at which a Type A's radius of curvature changes, lies a further two grid squares from the central axis, but falls exactly half way; along the vertical edge of a grid square. To achieve a grid in which these two key points would also be commensurate, the number of squares in the grid needs to be doubled to 28, or so it would seem. But in practice the metrology of a grid's side lengths would have ready made subdivisions, especially by half, and so one comes to the question of how early metrology defined units of length.
Figure 2 Some of the multiple squares present within the grid.
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In previous artices in this series it has been shown how disingenuous criticism of Alexander Thom's flattened circle geometries were, and how this has prevented further progress in understanding how they could have been (a) constructed, without ropes by using a grid, and (b) how the type-A for example can be organised using the multiple square geometries of megalithic Brittany (4700-3000BC). The location of useful astronomical geometries within a flattened circle, once drawn using squares rather than ropes, implies that flattened circles could have been a standard useful design for calculating and tracking important time periods. This idea can be taken further into the domain of observational astronomy, if such circles allowed observers to know which parts of the ecliptic are rising and setting on the horizon. In this regard, the circumpolar stars offer a natural timepiece rotating in the north, of a clear pattern of distinct stars and constellations: a design which is also circular.
In Sacred Number and the Lords of Time, I propose that circumpolar astronomy was practiced in the megalithic, from Brittany onwards, and this can account for the northerly alignments found within monuments, as being used to track sidereal time through either (a) direct viewing of the circumpolar region or (b) by noting the azimuth on the northern horizon of key (marker) stars which never set. The naked eye astronomer can then watch the circumpolar region and know the sidereal time through a direct experience even though there are distortions due to the angle of the ecliptic relative to the celestial equator, the ecliptic being skew to the polar axis and the circumpolar region being somewhat distorted by the act of reducing stars to their azimuth on the horizon.
With this in mind, another megalithic use for the Type A flattened circle can be imagined once one observes when the winter solstice point (on the eclipic) rises on the eastern horizon and when the summer solstice points rises, the time between is eight hours (in southern Brittany, 4300 BC). It then takes sixteen hours for the winter solstice point to again rise on the easter horizon. This observation can be done on winter solstice sunrise so that the sun will be setting eight hours later. Observations after summer solstice sunrise will offer a time (at Carnac) of sixteen hours. The generalisation that the length of the shortest winter day or longest summer day at a given latitude could tell the megalithic astronomer how to calibrate the circumpolar disc. In north-west Europe though, the days division of light and dark at the extremes of the year was very simple and the megalithic observer found the duration of the dark period (night) for one solstice equals the duration of the light period (day) for the opposite solstice. At latitudes near southern Brittany one also finds that the lesser period is 8 hours long and the longer period 16 hours as below.
The sun rises with the extreme northerly tip of the ecliptic at summer solstice and takes 16 hours before setting, at which point the most southerly tip of the ecliptic is rising on the horizon, and a night of 8 hours begins before the sun again appears still sitting near the northern extreme of the ecliptic. The Autumn equinoctal point, where the ecliptic drops below the celestial equator, rises at the midpoint between the solstitial points, and eight hours of earth rotation from each. The Spring equinoctal point sits similarly between the solstitial points, and four hours after the winter solstice point and four hours before the summer solstice point. This time sequence is forever slowly changing due to the progress of the sun along the equinox in (above) an anti-clockwise cycle through the year, with about 365 steps between positions at sunrise. This manes that any counting regime, of days around a circle representing a solar year, is naturally congruent to such a circumpolar understanding.
One can look at the circumpolar sky using software to see the actual pattern of stars that would have greeted the astronomers 6000 years ago, Firstly, one can see the circumpolar stars at summer solstice sunrise (or by symmetry, at winter solstice sunset):
image made using CyberSky
One is struck by the obvious fact that the Big Dipper is behaving like an hour hand for what we call the "two o'clock" angle of a modern clock's hour hand. In fact, the big dipper appears carved as a pattern of dots in the door jamb of La Table des Marchants, with an implied geometry of the 3-4-5 triangle, whose smaller angle defined the azimuth of solstices relative to east-west (sunrise or sunset).
If the midsummer sunset is observed in the northern sky at this epoch one sees the distinctive "question mark" down at the six o'clock position:
image made using CyberSky
The megalithic observer would therefore conceptualis the rotating pattern of stars according to the picture below, shown in the six o'clock position.
By looking at this timepiece, the parts of the ecliptic that were rising or setting on the horizon could be known, with particular focus on the four "gates" of the solar year, the solsticial and equinoctal points, in a pattern deduced from noting the dark periods of the two solstice days as in the ratio one third (summer) and two thirds (winter) of a complete (sidereal) day of one earth rotation. This same circle is the year circle of 365 days, reducing astronomy to a design naturally visible at the circumpolar region and which is elegantly presented in the Type-A flattened circle, a circle based upon the division into one third and two thirds.
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The diameter of Castle Rigg and hence the total side lengths of any grid of 14 squares, would be measured to be a whole number. to be 107.1 feet by Alexander Thom. The metrology of Egypt, where a grid of 14 squares is to be found evolved within the Rhind manuscript, made extensive use of the royal cubit, 1.5 times the royal foot of 8/7 feet and so 12/7 feet long. [*]
*This unit of length is explicit within stones C3 and R8 at Gavrinis and its derivation has been found by Robin Heath and myself to emerge directly from the use of megalithic yards to count time using one megalithic yard equal to a lunar month - the eclipse year then being one royal cubit less than the solar year.
If Castle Rigg was 108 feet in diameter, then this length is 63 royal cubits and each grid square would be 9/2 = 4.5 royal cubits long. However, one should at least expect the half royal cubit of 6/7 feet to have been available to the builders of Castle Rigg. This would enable the second pair of key points to be found from the first as each being two cubits (24/7 feet) further from the central axis and then being one and a half cubits (18/7 feet) downwards. If a foot is seen as containing twelve inches, then the royal cubit was quite likely to be seen, in its early usage, as also being made up of twelve units, each of 12/7 inch in length [*] and six such units making up 6/7 feet, the half cubit.
* In fact, within the colinear lines of the Gavrinis rock art one finds units of 12/7 inches between adjacent pecked lines and in the Kercado roof axe, a similar underlying unit of 12/7 inches.
All four key points at which the curvature changes within the Type A can therefore be seen to occur systematically within a grid of 14 by 14 squares. The arguments against Thom's hypothesis need to be reviewed since, within this grid, ropes could easily be used to find where stones should lie upon the regular arcs between these key points. However, arguments that stones were only placed approximately, by eye, also becomes achievable since one can easily imagine a quite accurate arc between two points, within such a grid. Meanwhile, the arguments that all such shapes were the result of inaccuracies when circles were intended, is an argument that seems to ignore the systematic conformity found within flattened circles in being symmetrical and flattened only on one side, usually only by about 85 to 90 percent of the circular diameter.