Is “Stecchini’s Cubit” The Draconic Megalithic Yard?
Even though it hasn’t yet helped to give a historical identity to the “Outer Sarsen Circle Unit”, I may get to say that looking at various units (where applicable) as things that can be constructed from whole numbers combined with exponential expressions of Pi has already proved helpful in several ways. It does seem to have helped us to identify the nature of the Draconic Megalithic Yard, a viable form of the Megalithic Yard that provides the most accurate expression of the number of days in the Draconic Month when its value is given in Imperial measures.
It also already allows us to make some very valuable inferences about the nature of Megalithic mesurements and what we might expect from certain Stone circle designs. At far right at Pi to the 4th and 5th powers we can where the “Outer Sarcen Diameter Unit” first came from, because with the “AE” Megalithic Yard of 2.72017497 – the primary unit of the outer sarsen circumference – constructed from Pi^5, we know its diameter will be constructed from (Pi^5)/Pi = (Pi^4) because of course circumference / Pi = diameter.
By the very same token we can see that if a circumference is in “Outer Sarsen Diameter Units” made from Pi^4, its diameter will then be the unit that utilizes Pi^3 (Pi^4/Pi) in combination with a whole number.
Thus we have more reason to expect that we should find some number of stone circles that have the “Outer Sarcen Diameter Unit” as the unit of their circumference, because this is categorically a way to write the remarkable approximation that is the Draconic Month / Draconic Megalithic Yard value into the diameter of the same circle.
For reference, the Draconic Megalithic Yard is suggested to measure
a calculated value of 27.21223280 / 10 feet, while the “textbook” value of the Draconic Month is 27.212220815 days.
For years now, I have been entering this neglected number into my calculator in the very same way I first discovered it, as (Generic Area of Circle 10313.24031 sq arc degrees OR Royal Cubit in Inches / 2) / ((Assyrian Cubit in Inches 19.46773764)^2) = 2.721223218).
In recent years, I’ve been finding out that some of the important equations formed by Stonehenge’s measurements are amenable to substitutions of Megalithic Yard values that are conducive to the discovery of this number in a manner that suggests its measures were very carefully and deliberately chosen in order to achieve this.
Almost twenty years ago, based on the dsecriptions from Stecchini’s text of three cubit rods described by Karl Lepsius, I tabled the possibility of a “Stecchini” or “Lepsius” Cubit. Michael Morton even in spite of the great pride he is entitled to as the first person to ever describe the Royal Cubit as it really seems to be, graciously entertained the possibility that I had discovered a valid secondary Royal Cubit value, which for a while I was referring to as the “B Version” of the Royal Cubit.
At the time, we did not know just how devoted the ancient Egyptians and others may have been to his geometrically standardized Royal Cubit value at the expense of all others. I have never needed a “Stecchini” Cubit to describe the exterior of a pyramid, and the only instance of it inside a pyramid that I’m certain of is one where “this is a very special case” is written all over it courtesy of other remarkably trend-breaking mathematics that goes with it.
(For those wanting to know more about that very unusual construct, it is the passage chamber of the Lahun pyramid recently discussed here.
Still, we wonder why such a Cubit Rod even exists if it isn’t an actual Cubit. I have been happy to follow along with some archaeological trends and refer to it as a “ceremonial” or “votive” Cubit value in that this may roughly correspond to how I think of it. The “Stecchini” Cubit may never have been considered a Royal Cubit at all and yet it would be advantageous to commemorate it with a measuring rod because there are times when we can use this value AS IF it were a Royal Cubit.
I would have to dig for it now, but I know that one scholarly article made reference to Petrie’s description of another probably so-called and equally “ceremonial” Cubit Rod that appears to measure 1.6 Hashimi Cubits. The distinction has caused me a bit of confusion before, and may even have caused a bit for researcher Geoff Bath as well.
Munck’s model of the Great Pyramid (“from the proposed pavement level”) is a prime example of this, which is something that initially caused confusion over which would have been the primary Egyptian Cubit and which of them was used in the Great Pyramid’s design, back when we were almost sure that this “Stecchini” Cubit could have been a genuine Royal Cubit; it also caused a bit of confusion over the proportions of the Great Pyramid’s missing apex section this way.
Essentially, even though we want to measure Munck’s model of the Great Pyramid in Morton’s Royal Cubit of 1.718873385 ft, once we have done this to recover its original proportions in that unit and hence in Imperial as well, then we can go back and remeasure it based on these original proportions, in a Cubit of 1.722570927 ft, in order to to recover additional important data (essentially, if we follow this lead, one thing it illuminates is the Egyptian Sacred Cubit), regardless of whether 1.722570927 was ever a genuine Royal Cubit or merely a mathematical constant similar to Morton’s Royal Cubit value.
This “Stecchini” Cubit drew some plausibility from being able to boast an origin and a standard in basic components of circular geometry, just as Morton’s Cubit could.
Morton’s Cubit can be constructed by squaring 360 and dividing by 2 Pi to obtain the Royal Cubit value in inches x 10^n; “Stecchini’s” Cubit can be constructed by halfing and inverting 360 then multiplying by Pi^3.
360 / 2 = 180; 1 / 180 = 5.5555555555 / 10^n; 5.5555555555 x (Pi^3) = 1.722570927 ft x 10^n.
For reference, the value was given by Stecchini as 525 millimeters
525.0000 mm = 1.722440945 ft
We may also wish to note that in addition to this Cubit Rod, Stecchini also called attention to a cubit rod of 524.1483 mm x .0003281 = 1.719735072 ft, which may either be Morton’s Cubit of 523.912607 mm or the incidental “Long Cubit” of 524.2915417 mm; another rod was cited of 528.3231 mm x .0003281 = 1.733428091 ft, which may be another “ceremonial” or honorary Cubit value, and may well be the honorary Cubit-like value that was included in Teti’s pyramidion as the true link between two very important unit values, even though it may never have been considered a Royal Cubit at all. (We now know that this cubit-like value is actually a measure in Stonehenge’s “Outer Sarsen Diameter Unit”).
I have stated in the past that we may occasionally see values of 1.722570927 or its simple multiples or fractions as ratios or products generated by architectural designs because it has some relatively minor astronomical significance, but I didn’t realize at the time that it was also intimately related to the Draconic Month. It appears there is still more to be learned about these things.
Now that we are learning more about metrological units that can be built on Pi exponents, the (Pi^3) in (Pi^3) = 1.722570927 tells us that this “Stecchini Cubit” should be a member of the Draconic Megalithic Yard family. I might never have guessed without taking a more analytical approach.
I’m curious to see what future discoveries might be made concerning a family of metrological units based on whole numbers and (Pi^3).
When Grasp Equaled Reach
In commenting on the palette of units on display at Stonehenge that seem most integral to its design, several times I have used the expression that “sometimes their reach exceeded their grasp” – that they aspired to show us a few things they didn’t actually have at their disposal, but almost did, because it comes with the territory.
One of them is that a side effect of combining the Remen and the Hashimi Cubit as major units of the same structure eventually results in a disappointment, and the Megalithic Foot, another major metrological unit of Stonehenge, is also involved in this.
Remen = 1.216733603 ft
Hashimi Cubit = 1.067438159 ft
Megalithic Foot = 1.177245771 ft
1.177245771 / 1.067438159 = 1.102870233 = sqrt 1.216322751
1.067438159^3 = 1.216264895
Thus in “Imperial”, our Great Universal Ancient Reference Unit, Remen / Megalithic Foot is almost the square root of the Remen, and the Hashimi Cubit is almost the cube root of the Remen – almost.
I presume the designers were aware of this, just as I’m aware of it, but decided to go ahead with this collection of units anyway, in spite of it being able to potentially cause some confusion or disappointment in just this way because of the great many benefits of such a combination.
Naturally we want to start tweaking these numbers, adjusting them slightly to make these “almost” equations come out true, but speaking from experience, it’s probably not just worth it. It may not even be mathematically possible. This may be about as close as we’re ever going to get. If anyone can demonstrate otherwise, please be my guest.
In fact, the real cube root and square root of the Remen don’t even belong to this system of numbers. The very system itself considers them invalid.
On the other hand, we should be careful and not be too hasty to think of the Remen-like numbers generated by Stonehenge’s “almost” equations as useless mathematical miscreants. Technically, because they are the product of multiplication operations involving two or more valid constants, they are valid numbers, and I have seen them rise to the surface just often enough in the course of experiments to know that however obscure they may be, they may have some important functions that have yet to be appreciated.
It’s also really not worth second guessing the particular primary unit values listed above for the Remen, Hashimi Cubit, and putatively the Megalithic Foot – they are time honored and true. I have been putting them to the test for the better part of 20 years now.
I recently had the good fortune of stumbling over several equally ambitious metrological equations that actually do worth, and then an hour ago while searching for something totally different (I was checking to see what’s been written on the Hipparchic Cycle – not nearly enough apparently), I stumbled across another older formula that I’ve been trying for several weeks to recover.
Here are the two more recent equations – the first one may not really be that recent a revelation, but I cannot yet locate any other history of its discovery besides the note I put to paper two days ago.
Megalithic Foot Cubed = .6 Incidental Megalithic Yards
1.177245771^3 = .6 x 2.719256444
Talk about something that makes me feel better about suggesting the crazy idea that a squared Megalithic Yard at Stonehenge is represented in root form as a pair of two different values, an “Alternate e’ Megalithic Yard (AEMY) and an “Incidental Megalithic Yard” (IMY) that together multiply to form the Squared Munck Megalithic Yard. It’s a bit bizarre and complicated both, but it is what the mathematics mandates, and more important, it actually works.
That is the nature of the squared Megalithic Yard, though – it is such that Munck’s value is what has the desired exponential value, even while its natural square root isn’t even valid according to the system it belongs to. “Squared Munck Megalithic Yard” is actually a misnomer because no true Megalithic Yard is involved in creating it as a square in the literal sense, but even though know we know it is 9 / Remen, no better descriptive has come along to date than “Squared Munck Megalithic Yard”.
Assyrian Cubit Cubed = 4 Hashimi Cubits
1.622311470^4 = 4 x 1.067438159
Here is the equation I have been several weeks trying to figure out where to look for. I posted about it at least as early as May 19 of 2020.
Squared Munck Megalithic Yard Cubed = (“Not-Phi” (short Assyrian Cubit) / 4) / 10^n
(2.719715671^2)^3 = 404.707285; 404.707285 x 4 = 1618.829140 = “Not-Phi” x 10^n
This is yet another fact that increases my confidence in the value of 1.618829140 as legitimate and intended. In spite of being able to say that this is a close adapation of a real Phi-like value common to every true pyramid with perimeter / height ratio of 2 Pi like some of those we find in Egypt, I realize that some people must find this number (and some of the places it has been found) bizarre enough that I still feel a bit apologetic about it. Suffice it to say it has proven itself time and again, and numerous architectural designs have been found that justify taking it to be second only to mighty 1.622311470 (the ordinary Assyrian Cubit) as Phi approximations go.
Recently we learned because of Stonehenge’s promoting, that as a metrological unit, “Not-Phi”, or the short Assyrian Cubit, is actually a form of the Hashimi Cubit, itself a form of the Egyptian Royal Foot. Some sources will make the Egyptian Royal Foot out to be a somewhat pointless 2/3 of a Royal Cubit, but the Great Pyramid’s chamber is one source that gives us the real story.
The often-seen diagram of the Pyramid of Niches at El Tajin (in Veracruz, Mexico), showing how the two most important ancient approximations of Phi appear as its diagonals. The missing figure at the right of the stairs at the bottom of the diagram is thought to be 11.1519204 ft, an expression in Egyptian Sacred Cubits, while the suspected figure at the left of the stairs is in Hashimi Cubits or Egyptian Royal Feet. The width at center is the most straightforward sort of expression possible of the Megalithic Foot.
1.618829140 even seems to have built in to an entire class of Thom’s Flattened Megalithic Rings, even though a subclass seems to be indicated; working with data from Ronald Curtis in Records in Stone: Papers in Memory of Alexander Thom it appears to me that at least several and probably a few flattened stone circles from this class may have successfully substituted 1.622311470 for 1.618829140.
Note the minor diameter / major diameter ratio of the Thom type B flattened ring (AB/BN contains a typographical error; it should read AB/MN like figures i and iii). We can take 0.8091 as half of Phi: 0.8091 x 2 = 1.6182 (1/2 Phi = 1.618033989 / 2 = 0.8090416994) but for years I have using 1.618829140 successfully with this design. With much respect for Phi proper, there is more to life and astronomical mathematics than Phi, such that it can be greatly advantageous to merely approximate it.
So there we have three different metrological formulas that are just as ambitious as some of those implied by the collection of units of measure prominently used at Stonehenge, but unlike the Remen root formulas, these metrological formulas work, and they work precisely.
An Emphatic Summary of Metrological Unit Relationships
Here is a more concise graphic summary of metrological unit relationships.
Sadly, three of these have only come to light for me in recent years. For years, I had only the concept of relating units though the geometry of squares or rectangles, as proposed by John Michell and others. It was only because of Geoff Bath’s timely emphasis on the concept of diametral vs circumferential relationships (the unit of the diameter of a circle vs the unit of the unit of its circumference).
It was timely because it’s the same consideration applied to circles and their measurements that we have with squares when we are talking about square based pyramids – if the Remen is sqrt 2 to the Royal Cubit and the base of a pyramid is measured in Royal Cubits, why shouldn’t its diagonal (sqrt 2 to the side of the base) be measured in Remens and not Royal Cubits?
I’ve been looking at pyramids that way for ages, but the same may not hold true for Egyptology.
With circles, this also holds true that the unit of the diameter shouldn’t be the same unit as the circumference. It’s a wonder I didn’t trip over this sooner because of Thom’s inclination to want both the diameter and the circumference of stone circles to be in Megalithic Yards.
What I didn’t realize is that so many of the unit values that I’d worked so hard on to figure what was their ideal value, were all interconnected, precisely, by the Pi ratio. You can imagine my shock last year to discover what had been right under my nose the whole time, because certain things had gotten in my way on the road to this conclusion, including that I was still thinking of some very important unit value as mathemaical constants, rather than as mathematical constants embodied by metrological units.
If one doesn’t believe that 1.177245771 or 1.067438159 is a metrological unit, it breaks the exquisite chain that is formed from ancient units through the geometry of circles, joining them together very much like beads on a strong. I would say that our bundle of ancient units must most definitely have a great deal of circular geometry in its lineage.
The “Swiss Army Knife” concept of ancient metrological unit relationships again. Our core group of ancient units and unit families are all “joined at the hip” like the tools in this pocket knife, in as many as FOUR different ways – some of them approximate, some of them exact.
Some readers may have already been following along last year as I began to take more and more notice of the way that the expanding collection of units were related to one another through multiplications and division because of a rapidly growing number of cases of architectural ratios between adjacent proportions in diverse units, whichwould spell out of the value of different units in “modern” “Imperial” feet as a ratio, which I was referring to as the “unit a times unit b” or “unit x times unit y” phenomenon (division is equally involved).
That started way back when I first realized that the missing section of the Great Pyramid related to the whole pyramid at a ratio of 10 Royal Cubits in feet, but it wasn’t until more recently when I began to avail myself more heavily of Petrie’s publications and data that were becoming more widely available, that this line of inquiry really began to build up steam.
–Luke Piwalker
Luke Piwalker 