The gematria of the semeia is a fairly modern approach, in that Euclids approached the subject using Epiphaneia.
I have benefited from years of additional thought and application, not the least being developing notions and terminology and etymological drift in word association. So it is natural that i would see the apriori notion of vector in the genatria of the semeion. However i am careful to distinguish the notion as a bicompass bi-vector system in the basic form. This is of course the basis of trigonometric land survey as well as geodesic surveyance. As far as i am concerned the basic vector notion was exploited in this way, and if Hipparchus takes the credit, then it is down to him, but i would of course point out that it is never that cut and dried, and Thales is reputed to have used such a system to calculate distances of "lengths". That may sound "funny" but one has to distinguish the subjective experience from the objective metron, and in a general phraseology to boot!
When i first tackled Monas i noted the kath" in kath'ekastoon, and subsequently derived etymoogically the "individual" notion in ekastoon, the singled out loner from a group. I later went on to derive etymologically the notion of katametria, the comparison by laying down a metron.
oi ontoi is interesting in itself, but the essential thing is that these are existing or real objects, they are objective therefore and not subjective, and interestingly can be "produced" that is manufacured, made sensible to the hand or the senses in general.
Thus to awake and see a connection between kath'ekaston and katametreesai is not surprising. The elasson and the meizonos if anything give a linguistic clue. The kat'ekaston is therefore a laid down singled out object or more directly a metron to be used in katmetreesai .
For individual beings …… ENUMERATION
The translation is treacherous! but it suffices to highlight the functional link. IN Euclid i do not expect enumeration i expect gematria and as for individual, when placed back into its context the phrase is defining monas as the case when an individual is "laid down" and called "one".
This individual object is a metron and it has form. The form is called arithmos partly because it is graspable "by the hand", and liftable into the air, and partly because it can be joined into a harmonia. From this aspect of the notion we get artois or harmonious fit and perissos which means fits around but not snuggly, just a bit too big!.http://en.wiktionary.org/wiki/ἀριθμός
The real objects that Euclid concentrates on initially, it is a teaching course after all , have a surface, an epiphaneia, This surface can be grasped and manipulated, and consequently it has "edges", perata. However Euclid is concentrating on a particular simbola to represent these edges and so he defines the gramme, introducing the use of drawing in the surface to define exactly what the topic focus is. Having defined gramme he focuses on semeion briefly. He does not define a sumbola for semeion.
Today we are so "hot" on the point, this seems an oversight. But in fact as i have explained at length, there was no need to define a symbol as any symbol for a subjective concept would do and in fact gramme cross producing the required referent. Semeia were not the focus of Euclids topics. Space was and is his focus.
Our first notion of space is therefore in the word "area", which of course is not a needed concept in Euclid, because the arithmoi are real forms. The concepts Euclid develops are meros and pollapleisios, which are derived forms . In deriving these forms Euclid develops the actions of division and multiples followed closely by the notion of sugkeimeia, that is aggregating to cover a form. The notion of subtraction is also inckuded in the development, but it is in the form of perissos , that is close enough, or approximation.
Thus Before any action is the form of space that governs or pertains to the development of that action. Losing sight of this is where the difficulty in mathematics arise. Thus for example , quite naturally, in tesselation with a triangle metron/monas we will find that the form to which the monas is being applied requires the monas to adapt different orientations.
Thus Before any action is the form of space that governs or pertains to the development of that action. Losing sight of this is where the difficulty in mathematics arise.
Think about this point carefully. I start with a large triangukar form. I decide the best way to "measure" that form is to use a smaller, similar triangle as a metron/monas. Why? Maybe i want to cmpare 2 triangular forms in terms of a common measure. This seems to be a common goal for comparison a notion fully utilised by Eudoxus.
Now i make the measurement by tiling. As i tile i notice that some of the triangle tiles have to be placed rotated π radians relative to others. Thus i do not have one metron but 2 which are related by a π radian rotation.
In common everyday use i would ignore this distinction and sum the tiles as one aggregate. This would give me a cipher for the monas, that is a scalar value for the metron. These scalar values would be related to triangle "dots" by the gematria of the Pythagoreans. Thus the Pythaagorean gematria dealt with a geometrical form in dots. It is enough to drive one dotty!(dots,by the way have become a standard symbol of a point, but not a semeion, which is now symbolised by a line vector)
However, when Descartes began to algebraise geometry, he did not include this notion of orientation, but rather the notion of imaginary. It was Bellini, and indeed prior to him Brahmagupta who introduced these spatial orientation relationships. Brahmagupta introduced them in the fullest sense, as balanced notions describing orientations in space signified in astrological and astronomical calculations. His relating of them to Shunya was to balance out the perceived error in the Monad philosophy of the greeks, which was influencing the traditional Brahman view of transformational, dynamic "origin".
Philosophically these differences are apparent only, as the reasoning behind the 2 philosophies is analogical. Cultural differences and insights however do distinguish the 2, and in particular the monadic modulus is more in line with Indian Philosophy of Shunya, that is "Fullness", than greek Philosophy of "Monad" that is "oneness or wholeness". The inherent notion in Shunya is "infinite", something the greeks tended to place "away" from their reasoning, as something which they could not reasonably attain to.
The greeks certainly considered the notion of the infinite, but as a process, Appolonius's famous stick process. but they could not conceive of it as a whole, that is as something essentially mensurable, from which commensurability may be defined.
Brahmagupta, and Indian and Chinese thought, subsequently modified by Buddhist Philosophy, vedic philosophy etc conceived of Brahma as whole, and the void from which the universe sprang as a transforming whole full of infinite potential and dynamic motion. thus we may see that the wholeneess idea unites all philosophies, but the kairos oor proportionality is where they differ. The Greeks defined proportionality through multiple. and essentially so do all philosophies, but Brahmagupta pointed out that muliples come in at least 2 forms, forms that cancel each other out. In fact they return the other to the infinite void, to the shunya of all things.
For Brahmagupta it was pertinent to advise, astrologically, on the fortunate ciphers and the unfortunate ciphers, that is scalars derived from the sifre which was the arabic word for Shunya from which we tended to down play the notion of infinite and play up the notion of nothingness or zero.. The reason is quite nationalistic! the western world did not want any thing to supersede the Greeks!
Bombelli by introducing his "pui" and "meno" began the process of freeing the negative numbers from their astrological gematrial associations, but the negative numbers have never really been liked since their introduction for this very reason: astrologically they mean doom and gloom!
However, Cardano and Tartaglia, introducing a notation for number from the Al Jibre did not understand the spatial significance of the "sign" of a number. Brahmagupta did however, but nobody, apart from possibly Bombelli wanted or could understand what he was advising. In addition rhetoric was the common mode of discourse, and it was Bombelli's innovative use of "signs", that is notation, that enasbled these distinctions to be isolated and pondered. Of course i do not Discount Vieta, nor the whole germanic rationalisation of notation and signage.
The point of importance is that the "sign" of a number has a spatial as well as a gematrial significance, and the spatial significance derives from the same notion from which i derive the notion of a compass vector network, the notion of the semeion.
The presentation of the difficulty as imaginary, or the √-1 has been a red herring for far too long. The fundamental difficulty has been in developing a consistent "vector" notion and notation based on space and gematrial considerations of monads.
We can now do that easily and consistently, and in fact we can see the connection between spherical Trigonometry and the development of a semiotic algebra(gematria) of vectors.
