Once again, the mathematics I am using uses a deliberately restricted pool of whole numbers.
Over time, the rationale for this has evolved somewhat. Fifteen or twenty years ago, if you had asked Michael Morton or I to explain this, we probably would have told you the reason for this concerned a desire for compatibility with sexigesimal math (the math that gives us 360 degrees to a circle, 60 minutes to an hour, sixty seconds to a minute, and so forth.
Since the discovery that Petrie’s “Inductive Metrology” apparently can be successfully applied after all to ancient architecture as a ground level analytical tool with only a slight expansion of the list of recognized metrological units, it also comes to light that even with this restricted set of whole numbers and this small number of ancient units of length measure, once the basic simple multiples and fractions are projected, there is already such a complex set of numbers generated that to not impose such stringent restrictions on the pool of acceptable whole numbers would generally result in unfathomable mathematical chaos.
One consequence of this restriction, is that it simultaneously loosens restrictions on the interpretation of excluded numbers; the rejected number 7 might be represented as 360/10^n Assyrian Cubits, or as ((1 / 45) x Pi) x 10^n and probably others.
In previous work on Stonehenge we see how this affects the “56” post holes of the Aubrey Circle. Since 56 is an excluded number, multiple approximations of 56 come into play there. Thus “56” is being used at Stonehenge as a “wildcard” with multiple correct interpretations, which hails back to the question of how we represent such decimal numbers as 55.89093145, which is probably the primary meaning of 56 of Stonehenge, through a number of architectural features.
This flexibility seems to correspond to the flexibility that may be required to use numbers near to 56 in astronomical equations.
Because of these, I have never tried to map out a list of likely primary approximations or definitive interpretations of excluded numbers, as there may not be any “definitive” interpretations; rather the interpretive value of indicated excluded numbers may depend on situation, context, and even some artistic licenses on the part of the architect.
As such, I really have no stock answer to the question of how we are to represent the excluded number 19, yet it is ultimately a question that needs to be dealt with to gain a better understanding of the representation of the Metonic Cycle. It’s probably fair to say that there is considerable mathematical evidence that the Metonic Cycle was known and used well before the time of Meton, its namesake who is generally credited with its discovery, yet the exact mathematics that is applicable has thus far remained somewhat elusive.
The great likelihood is that the ancient Maya were also quite aware of the Metonic Cycle and its utility, and yet they how they would have dealt with it and applied it exactly even in orthodox terms seems to also remain rather mysterious.
I’ve mentioned on a number of occasions that I don’t seem to have received much guidance from the ancients as to how to use approximations of 19, because every time they get up into that range, they seems to have some mandate to talk to about the Half Venus Cycle instead, and even that the Half Venus Cycle may often be what we are really seeing if we think we are seeing “19”.
So soon after the possible finding of some extremely noteworthy Aztec and Mayan calendar math in the cistern of Hadrian’s library complex, we can observe that while approximation of the Metonic Cycle as “19” Solar Years generally rests on the inference of the role of the canonical Solar Calendar Year, introduction of the Solar Year proper into this “Greek” formula tends to generate numbers resembling some of what would probably have to be considered the more controversial approximations of the Mayan Calendar Round.
So you can better see what I mean, let’s look at some equations
Textbook value” of Metonic Cycle (Wikipedia) 6939.688 days
6939.688 / 365 = 19.01284384 years
6939.688 / 365.25 = 18.99983025 years
6939 / 365 = 19.01095890 years
6939 / 365.25 = 18.99794661 years
“B” value for Half Venus Cycle = 18997.72194 days
The Metonic Cycle being a relatively large number, it is somewhat insensitive to small changes in input parameters in operations of division – it almost seems at first glance that there is no real distinction between “6939.688 / 365 = 19.01284384 years and 6939.688 / 365.25 = 18.99983025 years”, yet the same equation can also be very sensitive to changes in input parameters when seen as operation of multiplication. It may withstand that since we may be given some tolerance; representing 18980 as 18997.72194 adds some 17 days per approximately 19000 days, the maximum acceptable error or approximations thus far; proportionately, the figure of ~6939 would be allowed approximately +/-0 6.2 days, making the valid range for Metonic Cycle approximations to be about 6932.8 days-6945.2 days.
However, adding to the confusion is the fact we can generally find ways to approximate the Metonic Cycle with reasonably good accuracy so that we do not require such a wide tolerance. The cube of the reciprocal of 2 Remens, for example, has proven to be fairly pleasing and useful – and rather accurate – approximation of the Metonic Cycle.
From an ealier discussion,
“Step 1: Obtain the reciprocal of the Double Remen, as previously shown.
(1 / (1.216733603 x 2)) x 10 = 4.1093622960
Step 2: Multiply by 10
4.1093622960 x 10 = 41.093622960
Step 3: Cube this figure
((41.093622960 / 10)^3) = 6939.421817 x 10″
Mathematical probes have also returned some rather peculiar things regarding this formula of 19 Years = Metonic Cycle. I’m seeing at least one almost valid equation with an error ratio so low that I can’t recall seeing it ever before (making it that much more important that we work with tend digits).
This may prove to be related to the possibility of approximating the Metonic Cycle through a different formula:
((1 / Remen) x 10^n) / (12 x (Pi^2)) = 6939.425318
Where (12 x (Pi^2)) also represents the selected link between Venus Orbital Period and Half Venus Cycle, and which could therefore be conveniently recycled to facilitate a Metonic Cycle formula.
We might also recycle another number thought to be involved with calendar operations, 1.541011111 x 2 = 3.082022222, and multiply by the canonical Venus Orbital Period (“my” VOP B): 3.082022222 x 225 = 6934.55, which also falls within the projected tolerance range of 6932.8 days-6945.2 days. Currently, one candidate formula for the Eclipse Year is Venus Orbital Period x 1.541011111.
3.082022222 is thought to be encoded in an entire class of Thom’s flattened Megalithic rings; however it may be incompatable with Venus Orbital Figures for this particular purpose.
Likewise, we have another possible formula that may fail to deliver on promises of exactitude.
(365.0200808 / 192) x 365.0200808 = 6939.565593 / 10.
The cube of the reciprocal of 2 Remens nonetheless remains a promising possibility, including that recently we seem to have seen numerous examples from actual architecture showing the strong possibility that the ancients frequently referred to the Saros Cycle through a cube root figure as well.
One tempting possibility for a solution or alternative solution is that some branches of ancient astronomical mathematics may have opted to represent the Metonic Cycle simply by doubling the Eclipse Year: Eclipse Year 346.62 days x 2 = 6932.4, which depending on specifics might just pass for the lower end of the tolerance suggested for Metonic Cycle (min ~6932.8 days) that is proportional to the tolerance afforded to the Half Venus Cycle of necessity.
To do this with our best representation of the Eclipse Year, that is 346.5939350 x 2 x 10 = 6931.878703. Divided by the shorter calendar year of 3600 / Pi^2 = 300 Short Remens = 364.7562611 days, 6931.878703 / 364.7562611 = 19.00413904. However, 6931.878703 may prove to have little true affinity for more accurate representations of the Metonic Cycle.
For the record, 6931.878703 has already made an appearance as a projection in a set of experimental Metonic Cycle values based on the experimental formula, Saros x (1 / 1.053519238), which is some ways quite an attractive potential link between Saros and Metonic Cycle – yet the dynamics of this have managed to exclude what are otherwise probably the best candidates for the for Metonic Cycle.
Another experimental set places 6939.425316 in the “A” position and 6944.444444 ((1 / 144) x 10^n) in the “B” position, which also falls just within the proportional tolerance range of 6932.8 days-6945.2 days. This too may be worthy of some consideration.
Perhaps one other possible solution to at least some of this ((such as the larger errors) is that the Metonic Cycle was important enough to have been allowed two sets of values, which has seemed to prove the case for some of its possible components as well i.e., our mathematics hasn’t been forced to choose between different versions of calendar figures but instead has proven to be versatile enough to represent multiple versions of the Solar Year, Lunar Year and other figures that may merit such accommodation. It doesn’t seem that implausible that there were multiple forms of the Metonic Cycle, possibly depending on which version of the Solar Year we select as input (i.e., ~365 vs ~365.25).
If nothing else, this may be a good opportunity to review several points related to this.
In the early days of this work, it actually discouraged the idea that this math was about calendars in that the true Solar Year figure of about 365.25 days seems to extremely difficult for this mathematics to represent so accurately, believe it or not. Within the past year or so I discovered that it could be done – by dividing the Megalithic Foot by 12^n – but to do this requires such high exponential use of the number 12 that it will be rather difficult to justify its use in retrieval of the number, so essentially by the time we generate this more accurate Solar Year figure, it is already lost to us. For that, I’m not even going to mention what the number actually is, but anyone sufficiently curious can easily find out the answer to that for themselves given the clues.
What we seem to have been given by our numbers in the way of representation of 365.25 thus far tends to look more like 365.2767076 (133333.3333 / 365.0200808) and 365.2840914 (365.0200808 x 1.000723277).
(Note that Wikipdia informs us “Meton of Athens judged the cycle to be a whole number of days, 6,940” and that used in the canonical formula, 6940 / 19 = ~365.263).
Generally, on occasions when our equations give us something more like 365.25, it will be 365.2767076 or 365.2840914. At least one of these may be found in the Megalithic landscape as well as in Egypt; Alexander Thom’s Type A Flattened Ring possesses a “theta angle” of 19.10660535* which may have been intended as a square root of such a figure: 19.10660535^2 = 365.0623680. In terms of valid ways of representing this, sqrt 3.75 x (Pi^2) = 5 Egyptian Remens x Pi = 19.11240674 = sqrt 365.2840914.
19.11240674 is 1/16th of the calculated inner sarsen circle circumference of Stonehenge, and 1/32nd of the calculated slope length of the Great Pyramid from the base (and yes, that’s one the more obvious Stonehenge-Giza parallels of which there are a great many, that the Great Pyramid’s slope length works out to precisely twice the Sarcen Circle’s inner perimeter).
We might even be tempted for a moment to think that 19.10660535 could pass for “19” but again the multiplication operation is sensitive to variance, so that (sqrt 365) x 365 = 6973.315289, adding an unacceptable 34+ days to the Metonic Cycle – roughly twice the acceptable error even for the much larger Calendar Round.
To interpret these results so far as optimistically as possible, we may already have the very pieces we need to complete the picture; it may merely be a questions of finding an ordered way of organizing them.
Since I’m still not certain if the subject quite merits its own post, I will add here that our best value for the Eclipse Year also poses something of a metrological problem. Although, like many important numbers, it can be fashioned from established metrological units, in this case even the convergence of multiple metrological unit pairs on this particular Eclipse Year value does NOT seem to produce a quantity that can be readily identified as any of the 12 metrological unit families thus far.
Thus either something highly unusual is afoot here with the Eclipse Year value, or there may be a 13th family of unit values that has been overlooked.
I know that once I’ve made comment to GHMB about finding a metrological synonymy that was “masked”, which is so highly unusual as to be virtually unique so far, but I’m unable to find the information now. (I was certain I had specifically used the term “masked” but it has managed to evade search engines so far).
Could it be a case like that, or is it case of a still unidentified family of ancient units at work?
At any rate, even while it’s challenging to determine exactly what the ancients did about the “Metonic Problem”, we hopefully at least have what we need to see why it may not be unreasonable that the Metonic Cycle may have enjoyed universal use, well before the time of Meton of Athens.
If it takes 18 pages of calculations to do it, one of these days we will hopefully see the answer to the “Metonic Problem”.
In fact, before closing, I will check one more archival source for information about the Metonic Cycle that might have come from previous work, and I should also want to remind people of Jim Wakefield’s remarkable paper on the Rollright Circle, From the Rollrights to Stonehenge, in which he deftly developed a model in which both the Lunar Month and the Metronic Cycle are incorporated though the mathematics of a circle.
I haven’t felt the need to adapt Jim’s model although I’m aware of some possibilities of doing so, but this may also be a place to look for general guidance if the Metonic Cycle insists on being problematic.
This is from some of my notes which are dated to Jan 11, 2021 and are associated with experiments with projecting hexagonal geometry onto the Stonehenge model. While speculative in terms of where such a thing can actually be found in ancient architecture, it nonetheless identifies a notably well-integrated astronomical series.
“One reason that 43.60169520 may be attractive in spite of possibly being a relatively large departure from the raw data, is because of the way it responds to the all-important circular constant 2 Pi
When exposed to 2 Pi, it seems to becomes at very least a minor masterpiece of astronomical mathematics, forming a series at least as long as being bookended by both Metonic and Saros Cycle candidates, with other figures significant to astronomy also in the series
43.60169520 x (2 Pi)^1 = (1 / 365.0200808) x 10^n
43.60169520 x (2 Pi)^3 = Jupiter Orbital Period B / (4 x (10^n))
43.60169520 x (2 Pi)^4 = 6795.522395 (Lunar Nodal Precession 6793 days)
43.60169520 / (2 Pi)^1 = 6939.425318 / 10^n (best value to date for Metonic Cycle of 6939.688 days
This Metonic Cycle value is the cube of the reciprocal of 2 Remens.
(1 / (1.216733603 x 2))^3 = 6939.425318 / (10^5)
I want to emphasize that the “Lintel Circle Megalithic Yard”, 1/16th of 43.60169520 is simply a variant of the standard value I use for the Megalithic Foot
11.77245771 / (432 / 100) = 2.725105951 ft, Lintel Megalithic Yard
Therefore it seems perfectly accurate to suggest that the preceding display of astronomy and geometry magic in a series is brought to us courtesy the Harris-Stockdale Megalithic Foot.”
I also find a note reminding us that “Pi Jedi” have known 6939.425138 in its reciprocal form for quite some time now.
“144 x 1.000723277 = 144.10141520 might be preferable here because 1 / 144.10141520 = 6939.425313 / 10^n, most likely one of the better approximations of
There is also a note on how what is still ostensibly the best suggestion for the value of the Metonic Cycle has a geodetic connection.
“It also seems to be linked to geodesy: 6939.425322 / (12^3) = 1 / 24901.19742”.
I still don’t have any definitive answer, but that is hopefully most of the relevant observations concerning the Metonic Cycle to date, now all in one place to help offer guidance to any future attempts.
–Luke Piwalker
Luke Piwalker 