I’ve probably gone and got myself into trouble by attempting to be more comprehensive metrologically in my attempts at interpretation of some of the Callanish “stone circle” sites.
On the one hand, such an approach isn’t necessary, nor was it involved in many apparent interpretive successes – that includes interpreting the Great Pyramid interior, where a variety of measures had long been considered but were finally set aside to simply things long enough to finally succeed at a comfortable interpretation.
On the other hand, I consider myself a staunch supporter of the work of Harris and Stockdale and the validity of their work, and as such I’m increasingly eager to see how things look on conversion to Megalithic Feet and other ancient units, and whether we are heading toward consensus on some of the specifics of applied theories.
I can praise their work all day, but it will be a good question whether in real interpretive practice I actually achieve comparable results. What is more, the work of Harris and Stockdale on Callanish involves the Eclipse Year, which is not necessarily the easiest beast to wrestle with but rather the contrary, it may still be causing at least a small amount of confusion for many of us, including that David Kenworthy and others may have unnecessarily conflated it with the Royal Cubit.
It may merit repeating that my impression is that range of numbers near any metrological unit value can sometimes be tricky, not only because it only takes one word such as “geodesy” to turn a simple ancient unit of length measure into a maddening array of dubious unit values ala John Neal, but because when you work extensively with a particular unit of measure, you learn a lot of values which are near to that unit value but not equal.
From my perspective, the ancient Egyptians for having worked extensively with the Royal Cubit, quickly became aware of a lot of mathematical values that were near to it and similar to it, and even occasionally substituted one or another for the standard Royal Cubit value, even at considerable risk of confusing the would-be interpreter of ancient measure and architecture – which they apparently hoped to circumvent by providing context through specific combinations of numbers in a given setting.
Lest I digress, here is my hastily constructed table of data for Callanish, showing projections for Thom’s data in various units of measure that I believe to be associated with Stonehenge. I’m also going to break the table into pieces to make it easier to follow a more detailed discussion.
So far, there is no rule that says that a measure has to be meaningful no matter what unit measurement we convert it to. That’s something that’s still in experimental stages. It’s just as fair to say it suffices that the measurements are in the units we identify as fundamental to them — i.e, 40 Royal Cubits is something in itself and doesn’t also “need” to be a significant number of Remen or anything else – but multiple metrologies do create such possibilities, as does the transcendence of “Inductive Metrology” achieved by Stockdale and Harris, and myself.
In order words, a measurement doesn’t necessarily need to be a “nice round number” of any particular unit to be a fundamental value of that unit. A bit confusing? Sure — but not that confusing if even I can gain a grasp of it. A tiny bit of practice certainly helps a lot.
These are the specifications in feet that I decided on a long time ago for the Callanish I circle. At the time I had no idea what a “Pied du Roi” (Hashimi Cubit) or a Petrie Stonehenge Unit might be. I still stand by the figures in feet, because we can see it’s expeditious to also convert these measures in feet into other units.
Note that the circumference value in Pied du Rois (PDRs) comes out exactly equal to the value of 60 Megalithic Feet, which we seem to see at Callanish sites further down the list. Note that the value of 75.39822369 (24 Pi) feet in Petrie Stonehenge Units — 75.39822369 / (224.8373808 / 12) = 4.024147057 — is also data that we see metrologically at Stonehenge, where the mean diameter of the sarsen circle is proposed to be 100.6036764 ft — 100.6036764 x 4 =402.4147057.
The circumference value in Palestian Cubits (PAL) is data also contained in the pyramidion of the Great Pyramid according to my models (it’s the reciprocal of the pyramidion’s height).
So now we can see why such ordinarily “pedestrian” values as 24 feet might have been tolerated here — because to some degree, these figures are acting as “unifying values” (see Harris and Stockdale Astronomy and Measurement in Megalithic Architecture, page 28).
There is a somewhat strange thing here because of the specific combination of proportions, and that is that if we wish to insist on the validity of applying all of these units of measure, because of the curious properties of some of the numbers, the radial value of the Palestinian Cubit (PAL) may change to 2.106326983 ft even while the circumferential value does not, if we are to interpret 1.13903 as the square of the Hashimi Cubit / Pied du Roi = 1.067438159^2 = 1.139424223.
2.106326983 is a well pedigreed rather “Stonehengey” figure — 57.29577951 / 2.720174976 / 10 = 2.106326983, but as Palestinian Cubits go, it certain “plays second fiddle” to a Palestinian Cubit of 57.29577951 / 2.719256444 / 10 = 2.107038475.
Such occasional “juggling” of unit value sometimes happens – it is similar to and closely related to the “juggling” of Megalithic Yard values that Stonehenge was seemingly designed to capitalize on.
I’m also standing by the values I arrived at for the Callanish I Type A Flattened Ring long ago, which are 43.29292924 ft (Major Diameter), 39.47841760 ft (Pi^2 x 4) (Minor Diameter), and 132.3891319 ft — the “Callanish Number” (Perimeter).
Note that 39.47841760 also seems to behave as a “unifying value” – the value in Remens is 20 Assyrian Cubits in feet (39.47841760 / 1.216733603 = 1.622311470 x 20) and the value in Petrie Stonehenge Units is the optimum Palestian Cubit value (and vice versa) — that is, multiple combinations of metrological units point to this value.
Not content with that, the exact value in Megalithic Feet would be 39.47841760 / 1.177245771 = 33.53455886, which is 1/2 of 67.06911773, which has found its way into the last several previous posts despite their diverse subject material, and the value in Hashimi Cubits (PDRs) is 39.47841760 / 1.067438159 = 1/2 Squared Munck Megalithic Yard.
We may also wish to observe once again that 43.29292924 in Megalithic Feet is 43.29292924 / 1.177245771 = 1 / 2.719256444, the reciprocal or inverse form of the “Incidental” Megalithic Yard.
So far these have been good examples of the phenomenon where when we go to divide a piece of measurement data by one unit, we get a simple value of another unit in feet, which may have required deliberateness in design to achieve.
I’m still rather mystified by the perimeter value here, ~77.1486 ft. It’s quite likely too low to be Thom’s Mid Clyth Quantum.
Thom’s Mid Clyth Quantum as described in “Megalithic Sites in Britain”, pg 152
There may (or may not be) clues nearby. Note that the perimeter of the Callanish I Type A Ring — the “Callanish Number” — is in standard Royal Cubits 132.3891319 / 1.718873385 = 77.02087487, and the mysterious value of the perimeter of the larger ellipse of Callanish III in Palestian Cubits is approximately ~77.2739.
That’s something we see with considerable frequency at various sites, where the value of this measure in unit a is equal to the value that measure over there in unit b, something else that speaks of very careful and elaborate mathematical design.
I’ve underlined the value in Petrie Stonehenge Units (PSUs) because this is something in particular that looks like astronomical data, namely the duration in days of the Full Moon Cycle (FMC), although this remains to be seen. (The FMC is something of a strange creature, there are some outstanding candidates for primary values and yet a definite interpretation of the FMC is somehow still seemingly lacking).
I might have also underlined the value in Palestinian Cubits (PALs) because it looks a lot like a possible candidate for one of the rather elusive values for the Leap Year in days.
I find this curious because the area of the ellipse couldn’t be more obvious. Area value are interesting, typically useful, and sometimes decisive. It’s not always as easy to come up with a pleasing value for area in a diversity of units, which can have a limiting effect on the number of candidate units if both desirable and applicable.
The Squared Munck Megalithic Yard may be conditionally applicable here (just as we saw the Palestinian Cubit potentially being conditionally applicable ealier, back at the possibility of a different version of it in radius/diameter, and circumference, respectively).
For an area of 2 Pi, we obtain a valuable figure for applying the SMMY as the squared Megalithic Yard here, of sqrt 2160. (2 Pi) / 2.1719715671 = 46.47580015 sqrt 2160, and of course the SMMY being the Megalithic Yard that we can expect to have important exponential value, we can probably expect to a whole data series formed by combining it with 2 Pi (hint: the series includes the cubed Radian, 57.29577951^3, and the reciprocal of 1.177245771 – more Stonehenge data from Callanish).
One of the more important things that seems to be happening with this ellipse is that even with Thom’s suggestion of it being based on the simplest of Pythagorean triangles (the 3, 4,5 triangle), it’s a very good opportunity to see why “pedestrian” Megalithic Yard values are important – it’s because of their interactions with other ancient metrological units, something I finally learned from working with the Stonehenge ellipses.
What is very curious here is that the value of 3 Megalithic Yards, while important enough to be rather irresistible, is in standard Megalithic Feet about a week short of the Metonic Cycle of 6939.688 days. Perhaps such a figure belongs to an unusual set of considerations such as where a Lunar Leap Year is used in place of the sandard Lunar Year, or where ~224 is used in place of ~225 for the Venus Orbital Period, or where ~364 is used for the Solar Year or ~117 for the Mercury Synodic Period as the Maya (and apparently the architects of Stonehenge as well) were doing.
In many ways those variations are still largely unexplored territory (and you can see from my posting history just how recently those codex related variations have gotten onto the radar).
The conversion output for 3 Meg Yards of 2.720174976 is otherwise fairly stunning. In Remens, it’s this: (3 x 2.720174976) / 1.216733603 = 6.706911774, there is that yet again for at least the fourth time in only 3 posts. That’s 1.676727943 x 4.
In Hashimi Cubits/Pied du Rois, it’s (3 x 2.720174976)/ 1.067438159 = 7.644962672 — more “Stonehenge back-up data”: Inner sarsen circle circumference (360 / 1.177245771) ft / 4 = 76.44962693 = (sqrt 365.2840911) x 4.
In standard Palestinian Cubits, it’s (3 x 2.720174976) / 2.107038475 = 3.872983346 = sqrt 15 = proposed Mid Clyth Quantum value / 2.
Where’s all the astonomical data? It’s important to remember that astonomical data isn’t limited to Orbital Period or Synodic Cycles or another other cycles themselves per se, sometimes it’s ratios between those cycles for converting them for example.
Give or take decimal point placement, the Stonehenge inner sarcen circle circumference of 360 / 1.177245771 = 305.7985077 ft = 76.44962672 x 4 is not only confirmed as the functional ratio between the Orbital Periods of Venus and Mars (687.5493542 / 224.8373808 = 3.057985077, this same figure also still aspires to be the functional ratio between some form of the Lunar Year and the Mercury Synodic Period of ~115.88 days (354 / 115.88 = 3.054884363). That is astronomical data too, in addition to all the other functions those remarkable Stonehenge numbers serve.
Here we have the value of “26 Megalithic Yards” seemingly aspiring to be 60 standard Megalithic Feet (60 x 1.177245771 = 70.63474626 ft), and something again that is possibly suggestive of the Full Moon Cycle, appearing in the Royal Cubit column.
In the Palestinian Cubit (PAL) column, we what looks like still more of 33.53455886 = 67.06911772, although this is another instance that requires substituting a Palestianian Cubit of 57.29577951 / 2.720174976 for one of 57.29577951 / 2.719256444.
I’ve underlined the value of 2C = “13.9 MY” in Hashimi Cubits/Pied du Rois because of its resemblance to the ~354 day Lunar Year. To use the standard value (57.29577951 / 1.618829140) x 10 = Lunar Year 353.9334578, 353.9334578 x 1.067438159 = 377.8020786, the primary value approximating the ~378.09 day Saturn Synodic Period, linking these two being yet another of the astronomical functions of 1.067438159. The PDR value in feet is itself astronomical data.
Possibly the perimeter value in PALs, ~97.4638, is another clue – it’s certainly near to the Stonehenge inner sarsen circle diameter of 80 Remens = 80 x 1.216733603 = 97.33868824, although the question of which Palenestian Cubit (PAL) might have been used is yet to be considered.
For Callanish III, the circle diameter is presumably equal to the Major Diameter of the larger ellipse = 2A = 21 MY, the diameter therefor ebeing approximately 21 x 2.72 = 57.1200 ft x Pi = circumference 179.4477724. The meaning of this isn’t yet certain and it may not be impossible that the intended value is 57.29577951 ft diameter = 360/2 feet circumference – a 1/2 scale generic circle, as it were.
For 17 MY 46.24 / 2 = 23.12; 2.720174976 / 1.177245771 = 23.10626246. This makes the value of “17” the usual 20 / 1.177245771 = 16.98880598. In Remens, this becomes 1/2 of the C value of the Calendar Round / Half Venus Cycle, 18990.40381.
23.10626246 x 2 = 46.21252491; 46.21252491 / 1.216733603 = 3.798080763 = (18990.40381 x 2) / 10^n.
Notice especially the 2C value of “12.33 MY”, certainly a departure from any popular notions concerning Thom insisting on Integral numbers of Megalithic Yard. Much more importantly, we seem to find ourselves yet again looking at a value of 1.676727943 x 20 = 33.53455886, this time in “modern” feet.
The value in Megalithic Yards is therefore 33.53455886 / 1.177245771 = 28.48560571. This is 20 times 1.424280286, the Wonder Number first discovered at Tikal, then later found at Giza and at Stonehenge, not that surprisingly since a simple way to conjugate this number from familiar values is as yet another metrological “hybrid” 1.676727943 / 1.177245771 = 1.424280286.
In Remens, this seems to pass for the Anomalistic Month of ~27.55 — 33.53455886 / 1.216733603 = 27.56113481. This appears as the C value for the Anomalistic Month in the planetary cycle tables with 27.55182778 as the A value and 27.57175539 as the B value.
Notice that the value in PDRs here would of course be 10 Pi — 33.53455886 / 1.067438159 = 31.415926535. Since the value of the Perimeter in Megalithic Feet is suggestive of the Squared Megalithic Foot — 59.86 MY = 162.819 ft = ~138.305, we might speculate that what was intended could have been the Cubed Megalithic Foot
1.177245771^2 = 138.5907605 / 100 and 1.177245771^3 / 100 = 163.1553867, but for now that is merely speculation. After all they gave us quite a surprise at Stonehenge at with the bluestone “Oval With Corners”. The “Oval With Corners” number 138.6375741 could have quite easily been taken for 1.177245771^2 x 100 = 138.5907605 — in fact it nearly was — but it turned out to be, of all things, 1.177245771 x 1.177643424 = 138.6375741.
“1.177245771 x 1.177643424” simply isn’t the right way to look at it, as opposed to things like 1.067438159^2 x 1.216733603 = 138.6375741 that make far more sense of it. For that 138.6375741 may also be consdered something of a “unifying value” but more importantly it’s a simple multiple of the overall best representation of the Eclipse Year: (138.6375741 / 4) x 10 = 364.5939353.
In terms of unifying values, I was hoping to find the opportunity again somewhere in the course of this post to emphasize by way of specific example that we can also conjugate this Eclipse Year value from the 2.720174976 Megalithic Yard and 1.177245771 ft Megalithic Foot.
That is about all I feel certain enough about to discuss here for this item — hopefully, this has provided enough clues to make finishing the interpretation of the larger Callanish III ellipse more possible.
There’s our opportunity – 12.5 MY = 28.881 Megalithic Feet. With the numbers I use, when we see a number formatted a-b-b-b like that, there’s a good chance we may be looking at the Eclipse Year of ~346.62.
(12.5 x 2.720174976) / 1.177245771 = 28.88282807 = 346.5949468 / 12.
Be careful not to be taken in by this trick: 1 / 28.88282807 = 346.2264836 / 100, which produces an inferior approximation of the Eclipse Year that’s probably not worthy of the name.
I’m still willing to guess that the Perimeter value in feet may be 88.76223994 = SRVS, Square Root of the generic Volume of Sphere — as I have been for years now, but I should run the exact conversions to other units and check for exponents before I insist on that.
We may be able to take the 8.00 MY quite literally as 8 x 2.720174976. Note that in PDRs, this would be the same value as we get for the Callanish I circle in Megalithic Feet: 24 / 1.177245771 = 20.38656718.
The “9.6 MY” figure I”m not quite so certain of, but there may be a clue in that several times I thought I caught glimpses of the rarely discussed constant, Pi / 19.46773764 = 1.613743061, which probably doesn’t get enough publicity or usage because it’s not near enough to Phi the “Golden Ratio”, but it is a real constant, probably one with some unsuspected properties, and it may not be impossible that the intended value in PALs could be 2 / 1.613743061 = 2 / 12.39354670 for the raw value “12.3927”.
Finally in this set, we have the Callanish IV ellipse, at lower right above in Fig 11.3 from Thom, MSIB.
Caution: The perimeter value for the Callanish IV ellipse is mistated in both versions of the diagram appearin gon this page. Apparently a momentary bout with dyslexia, as the value given by Thom is actually 42.6 MY = 115.872.
I’ve underlined 34.650 for looking rather like the Eclipse Year in Remens, but how we get there exactly – or if we can – isn’t quite so obvious.
I should have also underlined “115.981” (not quite correct, the correct value should 115.872) for how much it resembles the textbook value for the Mercury Synodic Preiod of 115.88 days.
Wouldn’t it be wonderful if Callanish turns out to be a place we can go for guidance as to what the best value for the “MeSP” should actually be? That question managed to baffle me for years now (not that the Maya were helping by using 117 for it in their spare time).
Note that the value 42.160 looks rather like 20 standard Palestianian Cubits. 2.107038475 x 20 = 42.14076950, but 42.14076950 / 1.216733603 = 346.3434346 is thus far not an accepted primary value for the Eclipse Year. What may have been intended here is a shift to the secondary Remen value, which seems to very rarely used aside from it seemingly being a fact of life of life with Egyptian pyramid diagonals when their sides are in the standard Royal Cubits
1.718873385 x sqrt 2 = 1.215427027 x 2, closer to the perfected secondary value of 12 / (Pi^2) = 1.215854204 than the standard value of 1.216733603 ft. This substitution of an established Egyptian variation then gives us our overall best value for the Eclipse Year.
42.14076950 / 1.215854204 = 346.5939368 / 10.
For the value of 2C, we might begin with the observation that 1 / 28.2064 ft = 354.552043 and that we may be looking at some figure for the Lunar Year in reciprocal form.
The correct raw value of the Perimeter in Megalithic Feet should be 115.872 / 1.177245771 = 98.42464747, which looks rather like the reciprocal of 360 / Eclipse Year. It also resembles (2 / 1.622311470) x 8 = 98.62471107, which would make the value to be 98.62471107 x 1.177245771 = 116.1055240 for the suspected MeSP value.
Particularly with Mercury, it would be hard to guess which calendar group its values belong to to, but perhaps serendipitiously, (Solar Year A 364.7562611) / Pi = 11.61055240. If that lovely and very handy handy relationship were conserved as it probably should be, 11.61055240 x 10 would be the A value for the MeSP.
For ease of remebrance then, 116.1055240 is thus 3600 / (Pi^3).
Note that the raw value in Palestinan Cubits here is 55.043 — the whole bottom row is slightly incorrect, the raw value is actually 54.99282589, not that it matters much — which is suggestive of 50 Indus Feet in “modern” feet: 1.100874628 x 5 = 55.04373142, which is 1/10 of the base perimeter I’ve traditionally given for Silbury Hill for over a decade now.
Unfortunately, this may not actually all quite “add up” — keeping the MeSP figure of
For ease of remebrance then, 116.1055240 is thus 3600 / (Pi^3).
Note that the raw value in Palestinan Cubits here is 55.043 — the whole bottom row is slightly incorrect, the raw value is actually 54.99282589, not that it matters much — which is suggestive of 50 Indus Feet in “modern” feet: 1.100874628 x 5 = 55.04373142, which is 1/10 of the base perimeter I’ve traditionally given for Silbury Hill for over a decade now.
Unfortunately, this may not actually all quite “add up” — keeping the MeSP figure of
For ease of remebrance then, 116.1055240 is thus 3600 / (Pi^3).
Note that the raw value in Palestinan Cubits here is 55.043 — the whole bottom row is slightly incorrect, the raw value is actually 54.99282589, not that it matters much — which is suggestive of 50 Indus Feet in “modern” feet: 1.100874628 x 5 = 55.04373142, which is 1/10 of the base perimeter I’ve traditionally given for Silbury Hill for over a decade now.
Unfortunately, this may not actually all quite “add up” — keeping the MeSP figure of 116.1055240 and 55.04373142 would give a Palestinian Cubit of 2.109332361, and I’m not sure any of us are ready for that, so we may have to keep looking.
If we go back to the standard Palestian Cubit value of 2.107038475 feet
116.1055240 / 2.107038475 = 55.10365633
55.10365633 / 2 = 27.55182816, our best value for approximating the Anomalistic Month of 27.554551 days.
That might be a little more like what was intended here.
I think I will leave things right there for now and offer my thanks to readers for their kind patience if they’ve made it this far in this post.
Hopefully some hurdles have been cleared and some milestones achieved today in the understanding of Callanish sites I-IV and their metrological and mathematical relationships to Stonehenge.
–Luke Piwalker
Luke Piwalker 