[Some notes I made while I was writing up The Mathematical Origins of the Megalithic Yard in early 2020. The notes conclude with some observations of the importance of the idea of limit in Mesopotamia, and its connection with the Assyrian Sacred Tree, and their notion of kingship. I could have finished up with a short discussion of Egyptian interest in the idea of limit, particularly since we know (from the Rhind Papyrus) that they used the same method of calculation of Euler's number as in ancient Britain. That discussion with follow later.]
***
It has been twenty two days since I started to write up the
article ‘The Mathematical Origins of the Megalithic Yard’ (mid February 2020).
In this article, I suggested that those who designed the. circles came to the
idea of the megalithic yard of 2.72 feet as the consequence of an interest in
infinite series, and particularly those which approach a limit. The most
important of these limits is the one which is known as Euler’s number, which, when rounded up from 2.7218… is 2.72.
This limit was first noticed in relatively modern times in
the context of the calculation of compound interest, but the number, and the
process by which it is arrived at, can be found in many other contexts.
Effectively, the number (when worked out to thousands of
places), is the number as it would be found at infinity. So it can stand as an
indicator of ultimate limit and of infinity. It is associated with the idea of
‘one’, as I’ve discussed in the article, and also as an irrational equivalent
of one, which is a rational whole number.
An irrational counterpart to ‘one’, in a proto-pythagorean
community, would have been easy to understand as belonging to a world beyond
this one – i.e., a transcendent reality which is more perfect than this world,
which is full of irrationality and measures which are incommensurable. The
number may have been understood as being irrational to us because it is being
represented in our finite world, and not irrational.
It also stood for the edge of our reality, and therefore
would have signified the possibility of a joining between the transcendent
reality, and our world of physical reality. Finding ways in which the worlds
could be joined, and the incommensurate made commensurate, seems to have been a
major preoccupation in the Neolithic, as it was also to philosophers and
mathematicians in Greece during the second half of the first millennium BCE.
After I finished the article, I wondered how difficult it is
to construct a series which will arrive at Euler’s number, how it might have
been done, and how long it would take to come to the result.
A little research showed that there were many ways to
construct suitable series of numbers, and a geometric calculation could produce
a reasonable approximation reasonably quickly, without enormous calculations.
We don’t know for
certain what base was used for calculations in the British Neolithic, but they
were certainly aware of base 10, since they used powers of ten in their
construction (ie, instead of a 3,4,5 triangle, they would sometimes use 30,40,
50 as their measures, knowing that the sides would be similarly commensurate
after squaring). If they were using the English foot as their basic measure, it
is likely they were counting to base 12 (ie, in duodecimal). But the
construction of a series only requires whole numbers, arranged as fractions.
1 + 1/100000)^100000 =
2.7182682371923
100,000 is a lot of iterations, so it is unlikely that the
determination was done in this way. The process will result in Euler’s number
with any consistently generated series.
It can be done geometrically, which is much more practical,
and is probably the technique which was used in the Neolithic. Using a sequence
such as:
1/2 + 1/4
+ 1//8 +
1/16 + ... = 1
Those who generated such a geometrical figure did so knowing that the series converged on a limit from observing the initial results. What they wanted was to find out a reasonably accurate value for the limit itself. The square could therefore be of any size (read as the value ‘one’), and might well have been created in a large field, with the fractions indicated by small stones.
I’ve written elsewhere about the importance given to limits
and boundaries in ancient Assyria and Babylonia, particularly in connection
with sites connected with the gods, and the rituals for the installation of the
gods in Heaven. Sometimes aspects of the design of the Assyrian Sacred Tree
were unwrapped, and represented on pavings as lotuses, alternately open and
closed. Which is a way of indicating at these edge points that both
possibilities are open, and even perhaps that opposing states are commensurate
with each other in infinity.
It has already been identified that the Sacred Tree represents a form of limit, and consequently of the nature of divinity which has its true existence in a world beyond the constraints of finitude. The design of the alternating lotuses also was used to separate the registers of images adjoining the collosal Lamassu statues which guarded the entrances of royal palaces. There was an image of the sacred tree, with two winged genies behind Assurbanipal’s throne, which seems to indicate that the king was understood to embody the transcendent reality which lies behind the world of the here and now.[the identification of the king with the divine reality appears in various royal letters] He is the perfect man, and the very image of God
[March 8, 2020]
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