Sacred Number

After the last entry on Nineteen and the Lunar nodes, we should discuss the "lower register" of the Sun - Moon - Earth system, based on the solar year and the numbers 12 and 13.

Instead of 0.618 being attached to N [N=18 giving 18.618], 7/19 is attached (that is 0.36842) to N=12 to give 12.36842 lunar months per solar year, so that this fraction will resolve over the Metonic period of 19 years (see last posting). This is also quite closely proximate to 1/e (0.367879...), the reciprocal this time of the exponential constant 2.718281828...  rather than the golden mean.

19/7 is 2.7142857..., which whilst making it amenable to the Metonic also closely approximates the Megalithic yard of less than 2.72 - a unit of length recognised by Alexander Thom through statistical analysis of survey plans. One can see a relationship to the royal foot of 8/7 feet, its cubit of 12/7 feet and thence to a strange step of 19/7 feet rather than 20/7 [ a step usually being 2.5 feet). It must be the Astronomical Megalithic Yard that interests us here, a relevant formula for it being 19.008/7 feet, equally 19008/7000 which allows 19008 to provide prime number 11 in the numerator, and thence 11/7 making this a very fine Polar measure - it divides into the polar radius of the Earth, according to the ancient model of the Earth.

It was my brother's re-discovery of the Lunation Triangle with base 12, hypotenuse 13 and height 5 (the second Pythagorean triangle after 3-4-5), that revealed the ancient mechanism for harmonising the Sun and the Moon through an intermediate hypotenuse found at 3 high on the third side. Its length is the square root of 12² + 3² = 144 + 9 = 153 (fishes in a square root net, cast on the "right" side) 12.36931688... just slightly more than the 7/19 required for the Metonic.

This is the number of lunations in a solar year. It is close to but significantly not one third. The close thing gives you the rule of thumb, 37 lunations in three years, and interestingly 37 is a hexagonal number, 19 + 18. The number should be 365.242/29.53059 which equals 12.36825949, the underline being the signature of 7/19. The proximity to 1/e to 0.368 leads to the use of the megalithic yard (= e feet) to build the Lunation Triangle, whereupon the 0.368 approximation in the hypotenuse of 12.369, cancels with the 2.715428 feet approximation to e in the megalithic yard to yield exactly one foot at the end of 12 megalithic yards. The foot actually produced by this procedure is a Greek foot, 441/440 greater than the English but within that module.

This early link between metrology and astronomy through number appears very unlikely but was clearly intended and very useful.  It enables prediction of lunar phases and hence also of potential eclipses near the lunar nodes, also trackable using triangles (see previous) or using the Aubrey Hole simulator pioneered by Fred Hoyle and by inference, available as an early "clock" for astronomical events (see Robin Heath's descriptions of this).

I can now quote from an unpublished 1996 manuscript of a book written by Robin and myself.

Twelve and Thirteen

As we have seen, there are irrationals and their equivalent integers at work in the Moon's behaviour and it appears that the year is poised between the numbers twelve and thirteen, seen graphically in the lunation triangle.

Let us take the situation where the new moon began the year and twelve lunations [or lunar months] have passed. The Moon has returned to the Sun's position but 0.368 lunations remain in the year. As a part of the year, this period is 10.867 days long and therefore the Sun is 10.867 DAYS (of angle) from the year end on the Year Circle.

The Moon will travel 145.27 DAYS in that 0.368 lunations but 10.867 DAYS is travelled, by the Moon, in less than 20 hours and therefore [because the Moon reached year end in less than a day], in practical terms there are 13 orbits in [the time ]required for] 12 lunations. [In fact 12 lunations equals 12.9703 lunar orbits on average]

If we multiply 0.666 [2/3] by 12 over 13 we get 0.615 and so the ratio 12:13 is equivalent to 0.615/0.666 and, we think, a further veil has dropped away from the lunar number system. We can, in retrospect, see how the ratios equate like this: 0.666 is (2 over 3) which is (8 over 12/0.615) which is then 8/13 [because 8/13 = 0.615, a Fibonacci]

[If 12/13 = 0.615/0.666 = 8/13 times 3/2]

 The denominators are the creative forces and 12 evidently belongs to 0.666 [= 8/12] and 13 to 0.615 [=8/13], but in their relative size [one real fraction to the other], the other's denominator becomes the number of units relative to the other: [a fact perfectly expressed in the Lunation Triangle and ] a magnificent reciprocity based upon the field of pure number and expressed in the Moon's behaviour.

*** 

The square brackets are my attempt to make it more clear than mud. This remains an interesting insight linking again to a role for the Golden Mean, through Fibonacci approximation, in achieving near rationality within these two lunar time periods of orbit and synodic month. I went on (in Matrix of Creation) to find the reciprocal Golden Mean fraction of the lunation period, exactly 18.25 days, synchronous with Venus and the Practical Year of 365 days to a very high accuracy. The miracle is that all of these exact and approximate relationships coexist to create a rational environment at all, but this does require one to know the extent of the numerical network involved as well as its more sensational highlights.

To be able to form a hypothesis is a rare skill these days and this is exactly what is required to make sense of such an extended network of relationships.

"Knowledge gives nothing to a man until he gives everything to it"
(a Sufi aphorism through Idries Shah)