I’ve written before about the geometric relationships between proposed values for ancient units of measure. Ancient metrological units can be linked thought the geometry of both squares and rectangles, and the geometry of circles.
In terms of squares and rectangles, this is a concept I was introduced to by the work of John Michell, where we see direct relationship between the Egyptian Remen and other ancient established units of length measure through the classic square roots associated with and generated by the Vesica Piscis. Examples of this also occur in the work of Berriman, where we can observe the given Chinese “Ch’ih” value of 14.14 inches as 10 sqrt 2, making it (outrageously) the diagonal to the Imperial foot and a probable cognate of the Harris-Stockdale Megalithic Foot, and also in the works of WMF Petrie, where very simply, the diagonal of a square of 1 Royal Cubit per side, is the Remen (or the diagonal of a square of 1 Remen is 1/2 of a Royal Cubit).
In terms of relating units through circles, it would be one of the more important breakthroughs of 2020 that having followed Geoff Bath‘s lead on this, I was treated to not only resounding but absolutely unprecedented success after experimentally applying this concept. The understanding of ancient units seems to have grown by leaps and bounds thanks to Bath’s guidance.
In short, the geometry of circles allows us to link and organize ideal ancient unit values with absolute precision, whereas linking them through squares and rectangles requires slight approximation, giving us slightly different and somewhat flexible values for sqrt 2, sqrt 3, or sqrt 5. The approximations of these values that we would use are significant in themselves – for example, sqrt 2 is for intents and purposes the Harris-Stockdale Megalithic Yard in “Imperial” – but this set of relationships between units does not otherwise accommodate the possibility of exactitude, and may in fact try to automatically exclude it.
It occurred to me to write this post that some of what appear to be the primary ideal values for ancient units of measure can often also be connected to one another via themselves, and this phenomenon in general affords us with a set of rules regarding the interaction of unit values in combination.
Rather than trying to gather up and organize my notes so as to be able to compile some of the observations regarding this, I’m just going to try to conjure up a few random examples for the sake of illustration.
I should perhaps note that part of this might include the idea of metrological units as mathematical constants, meaning that beyond simply converting from one to another, we can apply some of these values in equations exponentially. Important data may be lost and design logic may be lost therefore if we limit ourselves to thinking of unit values as only that.
The other extreme end of the stick might be that if we become overly accustomed to thinking of unit values as mathematical constants with exponential data recovery value, it may become easier to overlook where import unit values may be authentic ancient units. Such has been the case not only with the Pied du Poi / Hashimi Cubit, which I was quite proficient at working with long before I ever dreamed it was a metrological unit, but also with the Harris-Stockdale Megalithic Foot.
Originally, making a metrological unit of the HSMF seemed like taking three steps backwards because as a mathematical constant, it’s particularly important. When I finally realized that like the Megalithic Yard, the HSMF is directly linked to ancient Egyptian units through geometry, from which (in my opinion) it draws considerable legitimacy as an ancient unit of measure.
These remarks about the power of the HSMF (or at least the number I use to represent it) also apply to mathematical relations between units.
For example, let’s take the standard primary Palestinian Cubit value of 2.107038473 ft, and divide it half: 2.107038473 / 2 = 1.053519237.
Now let’s multiply this by the value I use to represent the HSMF, which is synonymous with Carl Munck’s “Alternate Pi”: 1.177245771
1.053519237 x (1.177245771^1) = 1.240251066 = (4 x (Pi^3) / 10^n)
1.053519237 x (1.177245771^2) = 1.460080323 = 1.2 Remens of 1.216733603
1.053519237 x (1.177245771^3) = 1.718873385 = 1 Egyptian Royal Cubit
1.053519237 x (1.177245771^4) = 2.023536423 = 216/10^n inverse Hashimi Cubits
Thus some unit values can not only be linked in series through the ratio (2 Pi), but they can also sometimes be linked in series via one another, at least for those units we can expect to have good exponential value, such as the Megalithic Foot, the Squared Munck Megalithic Yard, the Remen, or the Thom Mid Clyth Quantum (~sqrt 60 ft).
Readers may recall a recent report here that the Thom Mid Clyth Quantum was discovered to be none other than the proposed Egyptian Sacred Cubit, whose proposed value is conjugated from what are probably the two most prominent ancient Egyptian Units, the Remen and the Royal Cubit. Very simply, Royal Cubit x Remen = Sacred Cubit, or to use my usual numbers, 1.718873385 x 1.216733603 = 2.091411007.
Some, such as Newton, may have conflated this with the Palestinian Cubit, which it is distinct from. (I in fact discovered this value while perusing Newton’s metrological writings and observing that the ratio between his Sacred Cubit and his Royal Cubit was about 1.213, highly suggestive of the Remen. I then went on to discover that this Sacred Cubit value of 2.091411007 ft has geodetic properties, and finally that it is synonymous with Thom’s Mid Clyth Quantum, which we have seen appearing persistently in architecture as diverse as that of ancient America and China.
The confirmation of this comes through squaring the Sacred Cubit and observing the square is a whole number, which is a property of unit values in the Mid Clyth Quantum series. 2.091411007^2 = 4374/1000.
That’s another example – and a very important one – of precise mathematical relationships between units.
We can also see therein another rule that belongs to any collection of such rules regarding unit relationships
Remen x Royal Cubit = Sacred Cubit / Mid Clyth Quantum
Here’s another example – this one may require more caution because we’ll use the Mid Clyth Quantum represented as sqrt 60 ft. In exponential use, of course every second application represents 60, so of course Royal Cubit x (Mid Clyth Quantum)^n where n is an odd number will also be in Royal Cubits. By the same virtue, all odd numbered powers will also give the same unit as output.
Nonetheless, even though it adds little to our collection of rules, the Mid Clyth Quantum can be used to align significant metrological series such as (correct decimal placement aside a moment)
1.718873385 / ((sqrt 60)^1) = 2.219055998 = 27/10 inverse Remens
1.718873385 / ((sqrt 60)^2) = Radian^2 / 10^n = 10^n x (Royal Cubit) / 6
1.718873385 / ((sqrt 60)^3) = 3.698426664 = 1/2 Squared Munck Meg Yard
1.718873385 / ((sqrt 60)^5) = 6.164044439 / 10 = 1 / Assyrian Cubit
1.718873385 / ((sqrt 60)^7) = 1.027340740 = 1 / (Roman-Egyptian Foot)
Thus it also serves to illustrate mathematical relationships between units and how they can lend themselves to aligning units in series, facilitating maximum data output in return for relatively minimal inquiry.
This is ideal because ideally every monument should acknowledge the possibility that is the first, and perhaps the only, monument we have analyzed. The architects presumably wish for important keys such as demonstrated here to leap out at us immediately so that we can understand them, rather than understanding requiring us to spend 20 years crunching the numbers from 40 different monuments.
In fact, the ancient habit of dropping clues is seen so often that there probably should be a name for it, although a good one continues to escape me (and the possibly synonymous “KISS” principle – “Keep It Simple, Stupid” is already well known). You can see it in the recent work I posted on Giza, where we find the Royal Cubit as ratio as many as three times – not 3 Royal Cubits, not 17 Royal Cubits, ONE Royal Cubit – it doesn’t get much simpler, or more obvious, than that). They clearly seem to be trying to bring us into things at “ground floor level” as much as possible.
Returning to the subject of unit relationship rules, let’s try to generate a few more examples of that for the sake of the discussion.
1 Royal Cubit 1.718873385 x 1 “LSR” Unit 1.676727943 = 2.882083035 ft = 27/10 Hashimi Cubits – so now we know the rule that Royal Cubits x LSRs = Hashimi Cubits.
1 Remen 1.216733603 x 1 “LSR” Unit 1.676727943 = 2.040131231 = 2.720174976 / 1.33333333 – so now we know the rule that Remens x LSRs = “AE” type Megalithic Yards (which we could also have inferred from 2.720174976 / 1.676727943 = 1.622311471, knowing as we do that an Assyrian Cubit of 1.622311471 ft = 4/3 of 1 Remen of 1.216733603).
Many other such observations are possible, such as 10.67438159 / 2.107038476 = 5.066059170 = 10.13211834 / 2, proving that “short” Greek Feet, and “short” Remens can be constructed from Hashimi Cubits and Palestinian Cubits.
It of course isn’t necessary to catalog such relationships or memorize them, yet clear a compilation of such relations would have the potential to remove a lot of guesswork, particularly from more metrologically-oriented analyses — and that is after all what I continue to experiment with, with much success, the premise that by applying Petrie’s Inductive Metrology, and by being aware of a small cluster of vital ancients units of measure, that no matter how complex the things that we can build from these various pieces, that the pieces themselves can generally be reduced to very simple metrological values.
Here are a couple of surprises though, from a page of notes I managed to locate, where the following formulas are listed as being true
AEMY Megalithic Yards / Sacred Cubits = Palestinian Cubits
IMY Megalithic Yard x Remens = AEMY Megalithic Yards.
There might be an important statement about the nature of the Incidental Megalithic Yard (IMY) in there that is going over my head?
It was also noted that values in LSR Units x Sacred Cubits = the diametral unit of the Stonehenge sarsen circle. This unit is poorly understood, but is the consequence of essentially accepting Thom’s proposal that the outer circumference of the sarsen circle should be 120 Megalithic Yards.
Thus, LSR Unit 1.676727943 x Sacred Cubit 2.091411007 = 3.506727276.
Fellow fans of Munck’s work may recognize that 360 x (Pi^4) = 35067.27277, but also, 350.6727276 / 216 = 1.623484851, which may be an oversized form of the Assyrian Cubit of 1.622311470 ft, but can definitely be seen (optionally) as a fundamental unit of the outer sarsen circle radius or diameter
1.623484851 x 32 = 51.9515151 = outer radius sarsen circle = outer circumference (120 x 2.720174976) / 2 Pi.
What 1.623484851 definitely isn’t, however, is 4/3 of the “short” Remen analogous to 1.622311470 / (4/3) = 1.216733603 – hence the lingering shroud of mystery.
–Luke Piwalker
Luke Piwalker 