Title: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 11, 2011, 07:51:25 AM The Logos Kairos Sumbola Sunthemata Summetria Theurgigical Response arises out of the visual, auditory gustatory kinaesthetic (proprioceptive) neural network interaction with the Shunya field. This neural network is maintained and enabled by the cellular microbiological interactions with the shunya field, maintaining a regional distinction between the biological field effect of the shunya field and the environmental field effect.
The Logos Kairos Sumbola Sunthemata Summetria Theurgigical Response is the main component of the subjective conscious process within the biological frame of the neural network, with the subjective unconscious processes arising within the biological structure as an evolutionarily determined outcome of shunya field interactions. On an environmental objective subjective description, the biological framework arises from the interaction between, and exists within a macro structure consisting of, similarly formed biological units. The macro structure is evolutionarily determined and environmentally bound within the shunya field. On a Shunya field perspective, the environmental boundaries for each subjectively distinct regional product within the field form a fractal distribution of the essential subfield structures of the multi-polar shunya field. This is a first draught of axiom 1 and it is a derived Pythagorean metaphysics. Title: Re: Axioms of the The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 22, 2011, 11:37:43 PM Axiom 2
The shunya field is a variable density driven rotational motion field, in which the multipolar structure forms the centres for the rotations of the spatial intensity. The spatial intensity is distributed according to the dual nature of the poles as either condensing or rarefying of space. The poles if they do collide merge to a uniform spatial intensity within the shunya field, but a uniform spatial intensity is not a equilibrium state, the shunya field is a dynamic equilibrium of poles. The spatial rotation is also dual, either it is one way or the other and is described relativistically by 4 states for any 2 poles, and in general 2^N states for N poles. The rotation is driven by the density dynamic of space as space condenses around some poles and rarefies around others ,but is spatially distributed in any 2 poles by the 4 states. The 4 states between any 2 poles doubles to 8 for the dual nature of the poles. Within these 8 states the rotational structures encapsulate the emergent properties of the Shunya field as subfield attributes. The Macro effect of the relative rotation multipolar field is an infinite iteration process and subprocess fractal structure that produces fractal Forms and dynamic iterative processes. Title: Re: Axioms of the The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 24, 2011, 02:05:23 AM Axiom3
The experiential continuum is a subjective experience that arises developmentally within the subjective process both consciously and unconsciously through the interaction with the shunya field iterative process. The earliest stage of the development is the formation of the experience of self , followed by the selection and amalgamation of the neural networks in the construction of "i" and "me". The interaction of the "i" and "me" with the interactive process within the iterative shunya field construct and develop the experiential continuum, as a model of objective space subjectively experienced with "me" in it. Self is an emergent subjective process experience that arises out of the interaction between a dense sensory boundary( sensorially dense), and a niche environment within the wider shunya field. The dense boundary is a fractal product of micro shunya field processes within a biological frame that is a macrofield fractal product, of a relatively local shunya field process structure. underpinning this development as an inherent scaffold is the logos summetria kairos sumbola sunthemata response. I sense, therefore I am. I am therefore i subjectively process objective sensory signals and output a model of my experience, that constitutes my experiential continuum. Title: Re: Axioms of the The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on January 13, 2012, 10:56:07 AM Axiom 4
My fundamental apprehension of space relies upon the assignment of semeia to regions of space, relative to my subjective processing centre, which provide a fundamental scatter pattern of semeia which may then be sequenced freely. Each sequence constitutes an output system status, and a synthesis or spatial relationships within the sequence. The complex of such systems i have synthesised under the notion of a network of vectors which dynamically vary orientation, in relationship to one another, and act as information conduits to the subjective processing centre. The dynamic information feed to the centre enables the system to be updated and the output to be changed iteratively. One essential output process is switching of relativity between the semeia by moving the apparent processing centre reference. This is a subjective and intriguing facility of the processing system, and is accomplished by iterative computation and reverse tracking of the information feeds in the sensory network systems. I establish the most effective model of these processes as the tools developed by The Grassmanns (Justus, Robert and Hermann) but principally organised in the Ausdehnungslehre. I acknowledge the debt these ideas owe to the Pythagorean school of Philosophy, and in particular the Fundamental Teaching Material of Euclid. Title: Re: Axioms of the The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 03, 2012, 10:38:00 AM Axiom 5: The Shunya Field in Axiom 1 is not constructed or definable within my model but is perceivable by "me" by a iterative process of negation of all elements within my model.
Basically i can't say what the Shunya Field is but i can say what it is not by recursive means. Axiom 6: I stabilise "my" model by an iterative process of "acceptance". "My" experiential continuum changes with what i "accept" as a basis for the iterative processes of perception and recognition. The cultural forms which i accept from conception are numerous and pervasive and as i alter these my perceptions change as does my experiential continuum. "not altering" then is a nascent notion of acceptance. Axiom 7: The Set FS is the "universal" set within which my model/ experiential continuum is defined and has a rule:- all processes on its elements are iterative /recursive and all its elements are determined by iterative processes. notFS will be the recursive definition of the Shunya Field in axiom 1. However there is a mapping from notFS onto FS such that FS is a model of notFS. Axiom 8: Iterative/recursive processes operating on notFS are perceivable. These processes will be compared with energetic transfomations within FS. Axiom 9:All perceived boundaries involve an iterative process or processes. Definition: Infinfite is unbounded and large Infinitesimal is unbounded and small. Euclid defined the term boundary and that is what i have used here without alteration. The axiom as i state it is a clarification of the definition of boundary and boundarisation. Axiom 7 has an interessting corollary. Energy and motion are by it recursive or iterative processes. This leads on reflection to the notion that a set wide iterative process may be a hypothesis worth making with regard to the notions of energy and motion in FS. This set wide process will only be worth making if the energy and motion laws that Einstein derived can be shown to be consistent by every measure with the axioms of the set FS. If this can be done then starting with a suitable fractal rule if recursive processes can be shown to generate Einstein like motions and energy equivalents then a convincing case may be made for the recursive action of space. If space has this recursive action it may then be possible to relate each iteration to a notion of sequential statuses which may be similar to the notion of "time" in modern physics. I will be more rigorous in a following post, but essentially the elements of the set FS are many and varied but for it to be useful FS must explain the nature of space or rather have an equivalent/analogous definition of space to what is in notFS. Axiom 7 requires some careful handling. Axioms 1 - 6 lay out an underpinning framework for 7 but do not define a set or set notation. This is in fact assumed to be the standard mathematical definition and usage. However, the axiom itself is attempting to draw together axioms 1 -6 under a mathematical notation system. Thus axiom 7 is a tautology expressing a symbolic representation (set FS with rule ) of axioms 1 - 6. Tautologies like this are indicative of the iterative nature of my consciousness, and the question arises if the generalised notion of iteration does not preclude me from coming to any other description. The definition of iteration is clearly too specific to explain everyday usage as I recognise iteration not only on bounded regions but also on values and symbols. Turings machine for example is a symbolic iteration, and Newtons iteration is a value iteration. Then there are the iterations which can be seen in design modification, Editing, scientific inductive reasoning, the scientific method itself acting on a group behavioural process; cybernetic and feedback systems etc. Therefore for FS to represent axioms 1 -6 the rule of iteration must take a more general form which can not rigorously be defined and is subjective to my appreciation of iteration. This is not a new problem. I mention it to highlight the fact that my so called knowledge of set FS is more likely to be a knowledge of a partial or restricted subset of FS or subsets of FS which may or may not be cofactors of one another. Title: Re: Axioms of the The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on April 01, 2012, 11:07:10 AM Axiom 10 i sketch out here as: large scale and small scale iteration procedures in FS are fractally entrained at any scale size i wish to examine.
When i take a large scale iterative process such as a solar cycle as a standard or the rotation of the earth about its axis, i can then subdivide it into smaller and smaller segments and use it as a measure or metric. Measures or metrics are some of the most obvious fractals that i can create or design in FS, but as such they are subjective attributes. Each iteration process i use as standard has to include the sub iterations within its "orbit" to imply any useful iteration link. So to become so "abstract" that a metric is applied "outside" of its defining iteration procedure is likely to lead to problems of scale. When i experience a fractal zoom it reminds me so strongly of the differences in structure which scale changes reveal and therefore it is a wisdom to me not to generalise in an assumption of a "smooth" continuous development beyond a certain iteration , procedures defining region of operation. Rather i should expect discontinuity and discrete regionalised developments. So for example the quantum physics and classical physics are fractally entrained by axiom 10 so they will operate on each other,but there will not be a smooth continuous link between them. The reason why Quantum and classical do not combine smoothly is because the number systems are different. Quantum uses an extension of the natural number system called Complex, or even quaternionic. There is no smooth transition between these two systems by design. They are adjugate or conjugate systems. The second issue is the scale of the forces. Gravity is often called space time curvature and a comparatively weak force toelectro magnetism. Thus it is approximately disjoint between these forces, as far as the data is concerned. Whatever descriptions we have of vorticular procedures should have a "fractal" nature if it is a "universally found" iteration, and boundary conditions will need to be generalised to reflect the wada nature of all boundaries in a fractal. So to follow on, a vortex process operating in an iterative way in set FS leads me to re emphasise newtons laws of motion in FS. A body continues in a state of vorticular motion iteration by iteration unless impressed upon by a force. A body impressed upon by a force changes its motion in proportion to the force and along the vorticular path of the force. And finally the impressed or drawn force is opposed by an equal and opposite force acting on the bodies involved . Newton accepted the states of motion and rest, but in set FS the iteration vortex is the source of all motion by fractal entrainment. A body is only at rest to an observer with the same vorticular motion, and i will discuss this more when i consider equilibrium and inertia. A body at rest is in a state of force equilibrium which is to say that all forces acting on the body cancel out. Only when this condition fails does a net force impress upon or draw upon a body in a "right" direction in the newtonian sense. This right direction is not a straight line but a perpendicular direction to the place of contact of the force. In a vortex field this approximates to a straight line as a first order approximation, but it is more accurately a logarithmic motion. Axiom11 Vorticular Iteration is the fractal entrainment underlying motion : a basic axiom of the set FS and a development of axiom 3 to be edited Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on May 28, 2013, 07:49:45 AM I have changed the lead title to reflect the metaphysical and philosophical nature of this thread. Perhaps it should now be moved to the Philosophy " channel" , Chris?
Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on July 14, 2013, 03:20:19 AM Recent research on Euclids Stoikeioon book 5 has developed my insight into Logos as a humanist response to the experiential continuum within and without.
As I have always said Logos is our response to the comparison of magnitudinal experiences. This is a ratio. But when I was taught raios it was always explained in terms of quantity. Numerals were then placed in front of me as symbols of quantity. No one explained magnitude to me. I had to figure that out myself 4 decades later! I book 5 a magnitude is placed in front of me. It is a straight line segment. Different magnitudes are apparent as different line segments. Mekos derives from mega, that is a general greatness. Mekothes is this general greatness as a description of a form, and it applies to any form open or closed, surface or stereoscopic. These are different kinds of magnitude. But then immediately , just as logos is defined it takes on the role of a model, a proto type which is used as an exact mould, something against which another logos is compared. Ana-logos, a new different logos in corresponding part to corresponding part, similar arrangement to similar arrangement. This exact copy is called an analogos even if the magnitudes are different, but only if the 2 logoi are equal or dual. The exactness of the copy is a subjective opinion. If the copy is almost exact this arrangement is formally defined by Benoit Mandelbrot as a fractal! Thus the notion of Fractal is based on Analogia, but ones where the analogos fails. The genius of Mandrlbrot's definition of almost self similarity defines intuitively analogos that is arbitrarily close to duality or equality. The analogos is a formal system. We subjectively sequence the form. The way to objectively see this is again based on the prior sequencing of an arbitrary mosaic. Within any mosaic, I subjectively identify four parts in a sequence. These four parts and their sequence are set in the mosaic! Now by using the language of distinctive order, protos, deuterons, tressaros, that is first, second, third, fourth I formally set those parts as positions in the language of rhetoric. Now actually on the mosaic part I may draw any line segment magnitude. If I do this for the four parts I have four line segment magnitudes in a formal sequence. The definition of a logos is the first and the second part compared. That means I compare the line magnitude drawn on the first part of the mosaic with the line magnitude of the second part of the mosaic. Already you see and feel the complexity of the rhetoric! The actual drawing could not be simpler! This sense of communication complexity is physically manifested. The subjective processing of the rhetoric causes a meditative or hypnagogic state of mind to enfold, while the eyes dart this way and that as each listened to word is visually and kinaesthetically processed. To an outside observer, this kind of response is simply called "thinking!". I call it internal and subjective processing. Hermann Grassmann called it Denken Akt. How would a blind person process this kind of mosaic based information? My answer is that such a person would have to process it kinaesthetically. Their proprioception would be entirely physical muscle memory, where each of the sequenced positions would have a muscle pattern say to the left and right. I mention the blind person because too often we ignore the contribution of our other senses to our "intuition". Also it helps to explain why rhetoric, or words are so slippery. If the word does not access the proper representational system, then the response of the processing is such that we call it " confusing". To overcome this, elementary rhetoric is strict, repetitive and highly referential. We do not find it pleasant to listen to this kind of rhetoric unless we are in "imprinting mode". Every newborn animate has a neurological set up which maximises information uptake, called imprinting. During the time of this set up most of the fundamental orientation learning takes place. What is learned at this time can help or hinder progress. Few realise that when they get to understand this process, they can go back and alter their imprinting. This involves meditative techniques which I will not go into here. So the mosaic pattern of four defines first a logos. The next 2 in the sequence define an ana logos!. Thus this is simply another logos, sequentially at the side of the protos logos, that is the first logos. But the relationship has to be fully defined to get the title Analogos. This definition is an algorithm. It defines an algorithm which essentially is a process of correspondence and comparison, after the second logos has been defined. I will not repeat the algorithm in this post, because I have written too much already! Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on July 27, 2013, 12:23:44 PM It is apparent that i should amend the heading of this Thread to the Logos Analogos... thus replacing Kairos.
But for the minute i will retain Kairos as the overall desired effectiveness of Astrological art, the advising of human souls on the opportune time at which to carry out their affairs and businesses! Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on August 02, 2013, 07:15:34 AM Research into the Eudoxian definitions show how this method has a specific framework set up. or algorithm. but that essentially it can be laid out in a pattern on the floor called a Sxesis, very similar to the word Sketch, or even Schematic. For me, this is yet further evidence of the imaginative use of the mosaic that the Pythagoreans made in their grasping into the nature of experiences.
http://jehovajah.wordpress.com/2013/08/02/eudoxus-on-logos/ Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on July 15, 2014, 12:27:57 PM There are several principles to our interaction with and inhabitation of space. At least I posit these are derivable from our intuitive Grammars which attempt to strap rules onto our unruly languages!
I will relate these to the 4 usual cases of the noun. There are more complex constructions of " cases" which I will ignore at this stage focussing only on the Nominative, accusative , dative and genitive cases. The first thing we consciously do, and it is an action or rather a reaction to space, is to name space or form or some form within a space or constituted by space. This is the nominative case and represents simple recognition or perception( even though these, Reactions are by definition iterative and fractal! ) The next stage of our reaction deals comprehensively with the relativity issues of space. The accusative case reflects the relativistic relation between two objects in dynamic relation. The simple accusative places the object relative to the subject in the activity. The object is either fixed or motive. Newton in his concept of relativity starts with a fixed or a uniformly moving object. This was done to provide a meaurable difference so that a system of measures could be derived. The accusative case is more complex and subtle than this however. Languages that retain the reflexive pronouns indicate the essential tautological nature of this case. Although English tries to straighten it out it cannot be straightened out without losing important properties and principles of relativity. We next move to comparisons of relative forms and this is usually the function of the dative case. The comparisons are fundamentally spatial! Thr prepositions are the basic distinctions in this spatial relativity, but also they form the basic reference Frame for meaning and analogy. The genitive case records how we flexibly adopt the relativistic viewpoint of another person or object. The aim is to experience what is fixed and what is motive from that position or point of view, in a dynamic situation.. The concept of solidity and fluidity is based firmly on this notion of fixity and motive. By adopting these viewpoints we compute a widespread notion of solid liquid and gas and plasma( fire!) . One thing we do not compute without deep thought is the concept of pressure and pressure gradients. These concepts determine the relative fixity and motion of space. We recognise this as relative " density" , but are mislead into thinking this intensity of fixedness or motion is somehow independent of relative viewpoint. Density is measured and apprehended by our kinaesthetic sensors. These represent a relative assessment of durability based on pressure interactions. What we usually think of as solid is in fact a highly local pressure system. Within a higher pressure system these systems become fluid. Within every solid system a fluid pressure system is in motion , and that motion is relative, not just to our conscious viewpoint, but also to objective reference frames. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on January 23, 2015, 09:53:29 AM The Axiomatic approach was a weird take on Euclidean philosophy so called but taken as a Platonic Philosophy rather than what it is a Pythagorean philosophy primer. Thus Kant based on some traditions and Legendres redaction of Euclids Stoikeia declared the postulates as Axioms. The Greek is not "axee " but "Aitema" which means demand. The word postulate means to demand or request, or beg.
Somehow the word got lost in translation and became equivalent to proposition! It would be more difficult to trace how that happened but essentially I assume that Aristitelian paradigms based on his logic of grammar and syllogistic forms was adopted first by Islamic scholars who then translated Euclid accordingly and mistranslated postulate as proposition. The result was that for nearly a millennia Arabic and Persian scholars attempted to derive the 5th postulate as if it were a proposition. This ultimately led to a crisis in Western European mathematics in the 1800s and the downgrading of Geometry, but just prior to that Kant and other Prussian philosophers had raised Axioms to a logical level of premise that cannot be demonstrated as true but are necessarily true. This is again an example of the profound difference between Aristotelian and Pythagorean thinking. The Pythagorean type of thinking is called rational, the Aristotelian is called Logical. One is based on Spatial interaction and the other one is based solely on grammar structures. On the face of it Aristotle had based his system on the " gatekeeper" position, because as far as he understood all thinking was done in and through language structures. Thus his taxonomy was derived by grammatical considerations , and his ontological categories was similarly derived. Rational thinking by Pythagoreans is based on spatial forms , but the system is reduced to ratios of line segments on a single line segment. This reduction is obtained by rotating line segments onto a given line, using circles. In fact one of the first propositions is about precisely this. This spatial and intuitive form of interacting with space engages the observer in what Grassmann called a real Expertise, whereas the Aristotelian logic only engages the observer in a Formal expertise! Why were the two " confused" for do long? The word Logis, a Greek Philosophical word obscured the 2 differing trains of thought For the Pythagoreans a logos was a description of a comparison, a ratio. The point being that in the presence of solely one thing nothing can be spoken or stated. A word begins in comparison . However Aristotle took Logos to be a word and the fundamental unit of language and grammar. He did not begin at the primitive nonverbal elements of space, but rather starts with words as primitive referents to objects and ideas.bthus his system was incapable of resolving beyond the formal definition of a word. The formal definition should admit more fundamental elements, but these are excluded by the insistence or the assumed equivalence of words to experiential objects. In the Pythagorean system the referent is not a word but on the text version a drawing, and in the course any number of physically sensible objects. Thus is revealed the basic structure of synthesis and analysis for real objects, the way real structures come into or go out of being in the observers experience and spatial location and at the moment of the observer reifying it. Aristotelian logic has difficulty in describing this experience and so calls them Axioms of their systems.. The two thought patterns thus do not fundamentally match so combining them leads to anomalous positions., so logic and Rayionality are in a tension that has to be managed, and this was best explored by Hegel who introduced a dilrcticsl procees based on comparing contrasting and concluding/ resolving. He later went on to redefine Logical Categories and ontological categories based on Kant's system but revised in the light of this dichotomy and Hegels rigorous and holistic analysis and synthesis. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on February 17, 2015, 08:27:56 AM The proto Indo European roots man* and me* underpin our Pre literate response to our interaction with space.
From a foetus we extend(metosis, meiosis) into space and that extension culminates in the limbs and the hand(manus) and the foot, and the tongue. These instruments probe , prod and explore space proprioceptively, developing a map as a copy of that probing experience. Thus me* denotes extension into and encompassing space, the proprioceptive intention of which is to " know" by comparing and contrasting results of the process. In that regard the hand plays a particular role ,man* denoting manipulating, holding, turning everyway and ultimately considering , drawing to ones side for special scrutiny , to come to conclusions. For these activities the Greeks coined the word Logos, but the idea is already present in the PIE roots and found in other linear B languages like phone in and Hebrew ( mamre). Given this fundamental extensive beginning one can trace counting as a consequence of this proceeding. Katametresee is bases on me* in combination with several other relevant notions, but fundamentally it is placing down and extensive body to cover a larger body . This is the act or activity of counting. It is literally a song and a dance: the dance rhythmically covers the form the song names each step. The names of each step only have an order if the Metron ( the extensive body used to measure) is set down in a sequence.. Where fors that sequence come from? It is an internal necessity of our development and growth, but it is also a derived process from natural processes. In this discussion no mention of points has arisen. That is because a point is a forml notion. Points do not have a real embodiment in some, although we can set out such a formal system. Points arise from our contrasting Anlysis of objects in spae, as we analyse an object it is dissolved into parts . When we arrive at a part that can no longer be divided then we call such a part a point. It is the first fundmental element of a synthesis of such elements by a sequence of construction. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on August 16, 2016, 12:17:15 PM It has been a while since I asked the question: where do sequences and series come from?
It is apparent that they arise from our human analytical and synthetical response to our interactions within space. Norman Wildberger had spent some time arcanely reinterpreting the foundational basis for a subject called mathematics to fit it for the computational age, but he has not tackled the origin question for these thought forms or " logical" structures. The word logic derives from Logos, but particularly in the context of methods of debating that " win" using logoi or symbolic patterns of expression. We do not often think of the words of our language as symbolic patterns because we interpret these patterns on the fly and not always accurately ! However it is usually the graphic symbols we utilise in astrological analysis and mosaic analysis. ( Arithmoi) The Stoikeion book 5 begins the instruction on how to analyse by means of Logos Analogos , a pattern of comparisons that highlights the duality of forms with different divisors / factors. Book 6 particularly takes this method to some sophistication, but it is in book7 that Eudoxus/ Euclid describes the Arithmoi constructed by the Euclidean highest common factor Algorithm. Book 7 forms a bridge between counting on line segments to counting using any type( genus) of magnitude . Again homogeneity as demanded in book 5 is crucial. The Algorithm reveals that the factoring process may have no natural end, and thus may not give a finite count. This was dealt with by declaring factors that had this result as protos, that is first in a new line of magnitudes! Proto Atithmoi often called prime numbers represent the deconstruction of magnitude into factors that are whole counts. Even so called irrational magnitudes are just re named prime magnitudes in the rational setting. It is from this factorising algorithm that we derive sequences of magnitudes by" Analysis" and in synthesis we construct "series" of these analytical elements to find the total count . Sequences and series arise from this analytical synthetical process of logos analogos comparison when factorising . http://m.youtube.com/watch?v=T_UAj92_hRY Unfortunately most mathematicians are used to putting multiplication preeminent to division . The Stoikeioon position is counting and factorising are the fundamental responses, the Logos .. Nevertheless this is an example of Normans systematic approach which makes sense. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on August 23, 2016, 10:53:56 AM The calculus is portrayed as an advanced subject accessible to those with prior learnings. Nevertheless it is a fundamental restatement of the basic process of factorising space, and spatial objects.
We may as well drop the term calculus and use the familiar term bricks . We may not all have played with bricks or in the sandpit when children but the logos response derives from there. The construction of forms we now give the grand name topology. When topology was first coined in the late 1950s the forms were morphable . This was in recognition to the overwhelming static nature of formal analysis hitherto. Classicists revered the Greeks codification of ideal forms and proportions based on aesthetic sensibilities. However we can see that different cultures had differing aesthetic values based on their sensibilities in particular thevindian Shunyasutras or trochoidal topologies, promulgated into Mongolia, China and Japan. So the mosaic patterns collected from the world cultures by the Arabic empires provides a rich topological set of bricks to choose from. Harriot, according to Wallis formulated a topological treatment of these forms and differences which the young genius DesCartes absorbed from the Jesuit scholars who instructed him. While plagiarism was not strictly defined at this time, it was a major concern among scholars who often obscured findings in coded descriptions to prevent others claiming authorship. With authorship came scholarly authority and that made a financial difference because expertise was valued and attracted patronage. And scholars urvived on patronage! Disciples or students are a meagre source of income but a wealthy patron could facilitate research and development of expertises for mutual benefit of the patron and scholar. In addition a scholars social standing was closely tied to their patrons. We therefore should not be surprised that scholars would imbibe each others ideas without giving due credit. DesCartes was notorious for this. Nevertheless his genius is not questioned even if his morals might be. DesCartes encouraged the study of differences . Differences here means differences in topology. Sticking to the Greek ideal forms often made topological geodesics difficult. In particular the curved forms of the eastern cultures were difficult to describe. However factorising the forms into smaller nd smaller regular nd recogniseble forms not only was a fractal process, but alo a differences process. Many series were devised by this method and the logarithmic and sine tables were derived by these differences methods expressed as calculation process formulae. The methods were expressed in a symbolic notation which stood for countable or measurable quantities. These expressions were later lumped under the term Algebra, by Wallis even though they had little to do with the Arabian Al Khwarzimi and his interest in finding the square and the roots of quadratic types of expressions . Bombelli is perhaps the earliest western author to publish on Alvebra as opposed to Arithmetic., but they constitute the same topic area, that is methods of calculating sums, areas and quantities and finding squares nd roots. The symbolic notation of Algdbra merely highlighted its status as ymbolic arithmetic. So the differences in algebra and the formulaic expressions of these difference processes became known by later students as differentials. These differentials were well studied by Newtons time but not well organised. We can trace two streams of organising this material to Newton And Leibniz, even in the face of the great works of the Bernoullis. The same core idea of factorising ino infinitesimal differences informed both, and both communicated with the others involved in the Mersenne correspondence community sharing scholarly ideas. However a difference in application and notation transpired between the two. I might briefly state it as Leibniz apprehended at last the principles of differential topology , the methods of calculating measurements of spatial forms. This is often caled differential geometry. However Newton apprehended the principles of differential dynamics ! There was no real equivalent to it . However that is not to say that Mechanics did not have all the required precursors. Mechnics is the application of topology to the methods of obstructing durable forms that can transmit and receive force or living force! These concepts of applying morphable forms to constructions Is not new, and in particular in understanding dynamics of motions or Kinematics morphable forms are the staring point. However, whereas Leibniz and others were studying the topological morphology, newtonwas studying the dynamic morphology. He was therefore studying forms that were Not static. To signify this he called his measures and symbols fluents. We may think the term amiable covers this, but it of ourse is too general. What istinguishesbabfluent from a variable is the time property of the magnitude. Up until then no one gave a Metron a time property. It was regarded as fixed.fixed by mental ecessity granted, but also later deemed to be fixed by the Gods or Musai. Newtons dynamical innovationnwasvto give all extensible and intenive magnitudes an inherent time property, that made them fluent. From this notion he derives the Fluxion, that is the small fluid change in topology over an infinitesimal Time change. Panta Rhei. All things flow with time! While this realisation is certainly obvious now, the pragmatics of scholarly analysis then denied this elemental philosophical truth. All scholars then looed for eternal fixed forms, even Leibniz. Only Newton looked for and expressed an eternal dynamic flow. Hence his terminology of fluents and Fluxions. In taxonomical terms classical geometry even differential Grometry is the geometry of a fluxion . All such geometry depicts an instantaneous topology , that is a topology at one instant in Time. Newtons method of fluxions thus deals with a dynamically flowing universe mom only called space-time . Space-time calculus originates with Newton and his method of Fluxions . Einsteins credit is to realise that nobody in his time had really read Newtons principles for Astologers or his papers on motion or the method of Fluxions. Those that had did not apprehend how to apply it to modern physics. He therefore redacted and resurrected it for his work on gravity. By thenn4 dimensional Algebras were common And Netons principles could be restated in these terms. The topology of space dynamically morphed could be equated to forces measured in space. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on January 02, 2017, 11:56:51 AM Xxx its been a while since I looked at the principles of so called Mathematics, the distinct learnings of the Pythagorean Mathematikoi or Astrologers.
What they did is ideate topological perceptions and dynamics . The idea, the foundation of ideation is the visual sense of any " thing" , or perception. Typically this is expressed as " an idea- usually a visual one, or a visible form. Thus a form in Grassmanns Formenlehre, and indeed in most of his peers minds was a thought Form, or an idea. Dedekind starts his analysis of number with this Greek idea . Quite apart from the misinformation taught about the Stoikeia and about the Arithmoi and about Monas , I find the discussion of Book 5, allegedly the work of Eudoxus, is not only muted but confused. Here I present a meditative translation of the first few lines of book 5. Here my initial identification of Skesis with the concept of a sketch is deepened by an etymological study of the word and particularly highlight the dichotomy with Skesis and kinesis, that is still and moving, static and dynamic, held and loosed to run free. Quote A Part is a magnitude of a magnitude , the lesser magnitude of the greater magnitude When Laid down onto it measures the greater Thus a Multiple Form is the greater form Made of the lesser forms When laid down it is measured by the lesser forms. Logos is two homogenous magnitudes Which apply to determining the stage of development of a specific make of sketch form Let Logos be Magnitudes" placed relative to each other" let's say, which generating magnitudes we "form into Multiple Forms " ( placed) relative to one another,letting be bigger and bigger magnitudes. http://perseus.uchicago.edu/cgi-bin/philologic/getobject.pl?c.31:3:35.LSJ And following entries. Peelikotees is an interrogative form, and therefore the noun form is nominalising a questioning activity. In the case of "the same logoi" magnitudes let be declared : the first placed near to the second and the third placed near to the fourth, when the magnitude of the first and the third is a dualled / duplicate multiple form , to the dualled/ duplicate multiple form of the second and fourth , That is : Applied onto the picked out one of the singled out ones applied onto any specifically numbered multiple form of it. And the both exceeds or the both duals or the both falls short Of the taken /picked one (when ) laid down together. Thus that case of "the same logos" magnitudes existing shall be called AnaLogos. [ the first statements of each book are definitions which are general. The general definitions make no sense without specific examples . The specific examples follow in the text and in point of fact are prior to the definitions! However the style is to pronounce the general form or principle drawn from experience like those given in the examples and then to give specific examples that reinforce and illustrate the general principle. Thus any student moves from not knowing to knowing terminology that is meaningless to attaching meaning to the terminology to appreciating the thought pattern being illicitrd. When this happens the student is initiated into the jargon of the senior level of discourse. A junior is not only evident by the use of unfamiliar terminology but also by the lack of ability to demonstrate examples of the terminology, and finally by not demonstrating the thought pattern used in the discourse. What if the thought pattern is " wrong" ? This can not be the case unless wrong means the pattern does not do what it is claimed it does. Thus if a logos is claimed to give eternal life then it is wrong but a logos is a particular pattern of 2 magnitudes which is used to generate multiple forms to determine the stage of development of a sketch of lines or flat shapes or solids , these latter 2 are dealt with in books 6 and 7 What I need to know : what are these dualled multiple forms What is Skesis ? What are the 2 homogenous magnitudes When are 2 logoi declared the same? What have to be taken together What are both and what do they exceed,dual or fall short of ? ] The first demonstration shows lines as magnitudes , it shows the generating magnitudes and it shows the multiple forms generated by dualed multiple forming . In this case the dual is the same number of magnitudes forming each respective. Multiple form Visually it is graspable, verbally it is convoluted! The ear , eye and thought are patterned to apprehend the same thing A logos is 2 magnitudes treated as a single pattern The relationships between magnitudes related in this way is what is being explored . The first demonstration is that multiple forms generated by the logos are dualled forms , that is the same or a dual number of magnitudes goes into each multiple form If you know the relationship of one then you know the relationship of the other because they are dualled. A logos is a pairing of any kind of magnitudes whatsoever? Strictly no , but the principles work for such Inhomogenous pairings. As I meditate on this I recognise that Monas is not a number . It is anything that we wish to call a unit , a whole entire. Even if it may be constructed from smaller units like Arithmoi generally are, we subsume those into a new Monas or unit. This is not the same as unity of parts, rather constituents of a unity may be inharmonious either in behaviour, aesthetics or shape, but we by force of will and convention, even convenience and custom declare them to be a unit, and thusly unify them!! The notion of a Metron is a secondary layer added to some topological form we clearly declare a unit. The Metron derives it's meaning from a song and dance process called katameetresee, in which we place down a unit while declaring or naming or singing a sound unique to that unit . But unique only in order in which we choose to proceed to sing ! Once laid down, the Arithmos is complete. But we may understand that this process of creating a perfect Arithmos require us to use a process of factorising. We always factor a larger by a smaller topological form . Thus in a process where forms are at first not commensurable, being either too many Or too little by a unit , that is perisos( as opposed to artios that is perfect fit) the Euclidean used the remainder to factor the whole thus they inverte the process by maintaining the rule: the smaller into the larger. This inversion continues until a perfect fit is found. That smallest remainder that does that thing is now the unit of commensurability for the 2 initial topological forms. But sometimes that remainder can not be found, and then the initial forms are said to be incommensurable. The ratios and proportions of book 5 explore these relationships between topological forms by analysing it in terms of extensible magnitudes. The layout for this exploration uses order( first second third and fourth etc) to keep track of the inversions in this process. From this layout, much later a Persian geometer constructed the rules for a system we call fractions. While fractions are purveyed as less than wholes, this is not the case. They are proportions and ratios of smaller units of one sort into larger units of another. Or more generally a lesser topological form into a greater topological form . The ratios and proportions remain the same as laid out in books 5 to 7 of the Stoikeia It is of great interest to note that from the outset these ratios and proportions deal with the dynamic process of stage development and how to depict it using recognised topological forms sketched in lines by the imagination of an artist ( Skesis) The etymology of Skesis, and linear B reveals the link to Aramaic, and the nearveasternnsemitic languages. In these languages the bilateral and trilateral root is important. The root "sk" root is associated with the hand and all it's functions . The main function is to hold The secondary function is with the hold to manipulate what is held, and the third aspect is to depict outcomes of the first 2 processes. Skazo means to slit by hand ( to loose something) Then the sk words up to skesis have to do with holding , retaining by being near to hand , being hand made, hand tooled , hand drawn , held in position, shape or form So ribs are like fingers holding in the viscera, and also holding the form The raft is a rough handmade form, binding together wood and Bouyant materials into a stable form . All of these ideas feed into the generl notion of a skesis. Thus a Skesis is a fixed form made by hand . Normally the hand element is way in the background , but in this context where the performer is doing a lot of creative processes, the hand element comes to the fore The idea of a sketch is seemingly tenuous, but it refers to the habit of geometers or rather Astrologers to sketch out the problem or situation either on paper or on the ground. It is apparent now that Verhältnisse is the German equivalent of Skesis, it being the everyway holded ness in space of some form or conception. The Logos and the Analogos. By the way I admit to formerly misinterpreting this section in the past. I hold my hand up as a hobbyist translator not a professional . Please challenge anything you like , but without ad homiem logic! Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on January 19, 2017, 06:01:58 AM http://m.youtube.com/watch?v=3grjQB04cu8
Here is aclear attempt to translate Euclids/ Eudoxus Logis Analogos theory. The definition is updated but clear. However it is brief and slightly misleading. Se the previous post for a fuller explanation The wot part is clear but the pattern magnitude of a magnitude is not expressed. Without this pattern homogeneity can not be explained. Two homogenous magnitudes are obtained from the same magnitude as parts. The process of making or creating multiple forms is not expressed. Instead the word multiple is used without reference to this process. The idea that a greater magnitude can be a multiple form created from a smaller form is not expressed. The reason is the difficulty of self reference in the Greek language. Thus the smaller form is a part of the larger form and the larger form is a multiple form created from the smaller form . But the smaller form is also part of its multiple form generated from itself by a process of dialling or duplication. Thus a multiple is formed or generated by a process of copying exactly or duplicating a form which is as a result a part of the multiple form and a lesser part of a greater magnitude or a lesser magnitude of a greater magnitude. Now this is the meditative expression of the ideas, pattern of words and content of the first discourse in the introduction. Despite its economy of words the text is not meant to be hurried over, but meditated upon to extract all the relationships and processes bound within it. The modern idea of an integer which means whole clearly clashes with the stated context of parts! And the reason how it does that is through the misleading concept of whole numbers rather than Arithmoi which actually are defined in book 7 . The process of multiplication is also misleading because there is no process of multiplication except by dual long or duplication of a given form or part. . The next idea is that of logos. Quite simply this is two parts from a larger form which are therefore homogenous. These parts can differ of be identical but the identical case is not used to define a logos. Two differing parts of the same form are used to define the word Logis! It is almost self evident that one can say very little about one form or one of anything. In fact the main thing said or say able about one is that is is the singled out form of singled out forms. Note the pattern again, established by magnitude of a magnitude. One therefore is left until book 7 to be fully defined where Monas is clearly defined as a singled out form of a singled out collection of forms. It is left until then because by then it is quite clear to the discoursing and meditating student that one is a very complex idea and not simple at all! Monas has an incredible discourse devoted to it by the Neopythagoreans. So the logos or the word is that two parts are laced next to each other in an unspecified way and those two parts are to be used to capture the development of a dynamic form . This dynamic form is developed by generating duplicates of the two initial or generating forms and making multiple forms which are therefore larger than or exceed the original homogenous pair of parts. These multiple forms are also placed next to each other . Consequently these multiple form pairs are also called logos, because the same word can be said about them. We now concentrate on the next idea which is Analogos again this is a pair of logoi considered together. The pair of logoi are referenced by their parts. The first and second part form one Logos and the third and fourth part form a second Logos We have just discussed how these logoi may be generated and the idea now is to define the same logoi by considering pairs of logoi. When we consider pairs of logoi this is defined as Analogos if and onl if the duplication of the first part gives the multiple form in the third part, and a duplicate duplication process on the second part gives the multiple firm in the fourth part. Analogos is thus about different logoi being declared the same or identical despite the fact that the parts are not the same! The two parts or both parts of the Analogos can be greater or exceed, or they can just dual, that is be an exact duplicate, or they can be less than the logos. the logos is thus parts the first and second, while ana( new or again) logos are partscthe third and fourth , and it is only thusly called if their is this duplicate duplicating process between the two or duplicated logoi! This heavy utilising of the duplication idea is an induction into the inductive reasoning and synthesis method of the pythagoreans. Because every statement of an idea or a process is duplicated at a different level, the next level up, the synthesis and arguments are irrefutable. And yet Atistotle attacked the process of synthesis on precisely this point. He tried to refute the notion of two or two ess or duality. I was not convinced by his refutation nor his argument against dual processes of synthesis or indeed against Pythagorean induction in general. It seemed to me rather that Aristotle , brilliant as he was, had do ewhat more to learn of the Pythagorean thought form, but he fixated more on Plato than on the teachings of the Pythagorean Mathdmatikoi of which Euclid and Eudoxys are both recognised members who have achieved that status. . Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on January 19, 2017, 07:26:01 AM The Ana Logos label is worth u dear standing a bit more.
http://m.youtube.com/watch?v=GnrCJiDXu2E First we have two parts of a form used to generate different multiple forms of those parts. The parts are paired in an unspecified manner exce T they are near to each other, and when they are paired in this way n expression is loosed out of the mouth and mind! Thus the simplest word, reference or description or comparison is two parts placed or conceived together. We could say "this pair" or" this relationship of two parts.". Whatefpver we say it is ubstantially more than we cnnsay of a part of a form. So the idea of defining or declaring two logoi as the same or identical or the self reveals the complexity of self referencing onthe Greek. The two logoi, the pair of logoi are declared self referencing when they actually reference each other! They do this when the multiple forms of the oarts of the logoi are generated from each other by a duplicating process the duplicates each part to a specified multiple form , and that specified multiple form is duplicated for the picked out or ingles out fist and third magnitudes and second and fourth magnitudes. So analogy is not based on omparing one homogenous magnitude or fom to mother, but on comparing a homogenous pair of forms to another pair which demonstrate a self referencing property. . We call this self reference similarity or proportion the Greek word is analogous. Two lines can not be of themselves analohous, even if one is generated as a multiple form of the other by duplicating pa part. The idea is to ompare - pir of parts and then compare another pair nd thn to determine if the logoi are nous. The next definition defines when one logos is greater than the second logo. It should be noted that the tutor refers to the confusing nature of the language . English avoids many self referencing verbs or ideas while it is a common feature of ncirnt near eastern languages. The simpler clearer language preferred by English speakers is actually less rich in implications and applications. . The meditative discourse is impoverished b these impolitic translations and the student is further disadvantaged by not being challenged to express clearly what they are experiencing and to sketch or draw their Ideas Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 01, 2017, 03:06:08 PM http://m.youtube.com/watch?v=9qEST41kG6Q
Somewhere in the Fractal foundations of mathematics I discuss Eulers equation for geometrical firms. V+F-E=2. This I related to points , lines surfaces and volumes to show that a kind of expression captures an invariant truth about rectilinear forms, and forms the basis for generalising into higher order or mor complex structures. This is the same idea explained here but presented as a general concept called a vector field. Norman Wildberger presents a powerful series showing Eulers formula to be topologically relevant , and especially when the lines are viewed as connections in a network. Here the topological constraint is area and volume. 2 surprising ideas are mentioned, dimension and orientation. http://m.youtube.com/watch?v=babedyM9dXg I believe Normans introduction to linear spaces is the most accessible general presentation. The concepts of span, linear combinations etc are carefully based on prior developments. The concepts of multisession, vexes, maxes etc all have clear if unfamiliar definitions. In addition, Norman does not shy away from the mathmythics and analogous thinking patterns that go into constructing a mathematical concept like dimension. I might add that I do not agree with some terminological niceties like linear instead of lineal, but the exposition is consonant with Grassmanns original ideas. So we can get an idea of dimension as a convenient way of synthesising the lines in a 3d space, a 2d space and 1d space. But then because we want to generalise from a specific case to analogous cases we drop the saciometrically interpretation and concentrate on the symbolic arithmetic, or Algebra Now Hermanns Grassmann recognised this as a articulately Hegelian dialectical approach , and his book, the Ausdehnungslehre is written from the general to the specific instances, as if this is how we think, hw we derive from these general thought patterns specific creative solutions. But Hermanns explains this is not the case. He came to this presentation after much trial and error , much inspiration that left major gaps that needed time to fill in, and he had so little time!! The story is declared in the thread The Theory of Stretchy Thingys!! I mention it because few find the purity of this presentation( Fōrderung) easy to deal with. And the concept of orientation, a fundamental spaciometric concept comes in right at the beginning after a general combinatorics discussion requires it. Norman introduces right at the beginning as a property of the natural numbers when written as multisets. The arrangement of marks on the page demand sequence, spatial orientation , structural form etc.. As spaciometric entities, our proprioceotion demands the concept of orientation . But here s the rub: Hermanns used the concept of direction rather than pure orientation, here Macdonald uses a non spatially specific concept of orientation except in the scalers where he has to specify an order relation as an orientation. Hermanns took his primitives to be continuos line segments,Strecken, and thus properties of orientation, direction and magnitude and quantum naturally follow, or are inherent in the primitive. However in his presentation he starts with the undefined primitive of a dynamic point.there is an interplay between this creative dynamic primitive and the static reference point defined using it in a tautological statement early on in his presentation.. It is this dynamic creative point that gives ' direction' through its travel to a created concept called a line. The line is created by this dynamic primitive entity. In "reality" it is a fancy for our mutable perceptive abilities, in that I may perceive such a point as dynamically creating a path called a line, even where no such physical dynamic palpably exists. So in this regard, orientation is used to invoke the direction of this creative point, and therefore it fails for a solid , returning to a single continuous line that traverses throughout the space! The dimension of 1 for a solid is because one set of vectors describes every reference point. We could therefore creat every reference poit by a continuous line that travels to and through every point. Hilbert showed that we can logically come to this conclusion by a limit argument. Of course you need to be convinced such infinite actions make sense! But it is possible to do this fr any finite set of rational reference points.it would have to be by this line that " orientation" of a solid is defined, and as a starting point is arbitrary, so will any orientation be! The use of rotation to define orientation sets up some interesting problems. Essentially we could spiral around a diagonal like an onion in layers, and that reveals the essential rotational dy amic we absolutely need to make sense of our space! It was this freedom of rotation that baffled Hermanns, but which drove Hamilton to a brute force solution for 3 dimensions we call the Quaternions. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 04, 2017, 01:34:34 PM It dawned on me during a bus ride that linear ombinations, better general combinations, are not restricted to oriented li e segments . In fact I was thinking about the minimum basis to reference motion, and realised that 3 dimensions are fine for point referenced motion along some line, but motion in a general p,and requires some serious constraints.
I could use on st of orthogonal reference frames to give position. Then a different set to give rotation, consisting of trigonometric lines. But the rotation of a sphere is not reducible to one set of orthogonal trigonometric linesc because those lines would be oriented in the position frame and give a different behaviour for each orientation. The orientation does not affect the combination, only the terms combined, and those terms are not reducible to others by combination . So in general to reference motion we need as many orthogonal trigonometric sets as necessary to describe the motion. This would be a general combination and so we would have at least 9 dimensions for any motion and probably more ! Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 06, 2017, 12:19:24 PM http://m.youtube.com/watch?v=8vBfTyBPu-4
I thought I might try to grasp the notions of co and contravariance. http://m.youtube.com/watch?v=AKPZkHvqTao As usual, the mathmythics fogs up the view. First consider Thales theorem ; in a semi circle one always sub tends a right angle at the perimeter. . The straight lines that are subtending, that is holding below, actually hold the diameter below the semi circular perimeter, from a point on it. These lines are the sides or limbs of a right angled triangle that stands on the diameter. So now consider a full circle crossed by a diameter. I can find 2 points one in either semi perimeter such that the lines at one end of the diameter are at a given angle between them . Such a pair of lines may be taken as the reference frame for a plane coordinate system. When such a pair of lines are used they clearly vary by the perimeter of this circle, and the diameter can be seen to project orthogonally onto these lines in each semi circle from the opposing end. This dynamic is called co variance of the reference frame lines. The lines do not in fact vary together as co implies, they vary independently depending on the desire of the observer. So the term is immediately misleading. The lines are on strained by the diameter and the perimeter of the circle to relate by Thales theorem However, if we concentrate on the fact the vertical projection onto the reference lines ar both the diameter and a Cosine product then we could call them covariants, either as a shortened " Cosine variant" or a co variant as in fellow type or kind of variant of the diameter Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 06, 2017, 09:22:32 PM The contra variant uses parallel projection as the resolving process. In this case the controlling figure is a arallelogram with a fixed diameter while the other diameter is allowed to freely extend as the lines remain parallel.
We may take one end of this fixed diameter as the place to fix a reference frame. In this case the parallel projection onto these lines now vary in opposing ways with the angle between them . In this case the projected lines cut off varying magnitudes of the reference lines, but in a way that is co varying, dependent on each other by the angle between them! The contra idea comes only from the observation that they extend in an opposing manner while not truly opposite extensions. The introduction of the tensor matrix us interesting. These change if basis processes can be encapsulated in an array. The array sets out systematically on the page the factors of a product. The idea of setting the process out in this way is ascribed among others to Cayley, but in fact Grassmann spent most of the Ausdehnunslehre specially setting out these product sums. The array or aerie was key to his conception even if he did not create a new tabular presentation of the factors. It is this tabular arrangement that was at first called a table and then a Matrix, and to a certain extent zest Vainant contributed to this key to understanding the then modern mathematical processes. We must however understand that it was Peano who took Hermanns Grassmans work to the format we mysteriously cLlmTensirs! There is no mystery, the Italian Tensor means to stretch just as the German Ausdehnungs does. So the Tensor theory relied heavily on the summation convention or the element with subscripts for row and column inventions. . The Cayley table format gave a specific table to wrk with rather than just a symbolic element with row column identifiers. Certain products could be easily distinguished by this notational device, however, and whole matrices could be referred to elegantly. The distinction between circular based referee frames and parallelogram based reference frames became so useful that the superscript notation was devised by Ricci and Levi, and taught to Einstein. These arrays were identified as algebraic in themselves and this enabled Ricci to depict complex combinations of terms to produce products very elegantly. The matrix that enabled certain equations to be expressed legantly providing they were affine projections or circular projections were called by the Ricci Levi school Tensors. They represented 2 types of extension or extensive magnitudes; the trigonometric and the affine. The projective extension came later and possibly the most accessible proponent of that is NJ Wildberger. http://m.youtube.com/watch?v=6MstJEAlNFs Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 08, 2017, 12:43:06 PM I have been attempting to understand trochoidal dynamics.
Hang on , I have tried to expand my thinking to free it from the tyranny of the straight line! Forgive the poetry. I have struggled for years to apprehend what mathematics and mathematical physics and chemistry is about. Finally I accepted thebGrassmanns approach, a constructivist approach. . To cut a long introduction short searching for Grassmann in YouTube brought me into contact with NJWildberger. Now I have used his research nd understanding to better grasp the Grassmanns group and ring theoretical combinatorial approach. But then it became clear they were expounding on the Pythagorean school of thought. My interest was then piqued by the fundamentally dynamic basis provided by the circle or sphere. Then I found out that Örsted had found experimental evidence for a circular force dynamic being inherent in space. Then I recently found the Boscovich exposition of inherent force expounding on Newtonial principles. So the Benoit Mandelbrot Fractal topology revolution gave me a computational dynamic for rotational dynamics . Later I came across Kegan J Brills trochoidal spiral surfaces and William Shank and his Trochoidal applications along with TerrybGintz fractal generators. Some original research I to the imaginaries convinced me that the mystery was man made. The quarter arc magnitude , while topologically familiar was void in the algebraic or symbolic arithmetic setting. It has taken a while to grasp how a topological quarter arc is combined on the page, in space and in space-time. So I found a Will Shank app called TroTorted very suggestive for representing a Boscovichian pressure dynamic. I got excited enough to try to depict the Hydrogen datac. This revealed that Trotorted was limited by its design Finally I was driven to another Will Shank app called Circa. , which was clearly just planar. However the addition of a dynamic rotation on screen meant that visually at least I could trick the eye into appreciating a 3d aspect to the trochoidal forms. In coming to this arrangement of the phase scrubbers in Circa I realised that it was an implementation of NUWildbergers ISpan technology, with dynamics added. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 08, 2017, 01:08:31 PM http://m.youtube.com/watch?=SZmlo49D1NY
The Mset data set basis to integers or whole form conceptions While the anti mark is useful it is better to think of a mirror form While a mirror image has no reality a mirror form is a partial negative that cancels out part of the real form. Thus perpendicular reflected dimensions are negatives but those that are arallelogram to the mirror are unaffected . The best negative is the hole that snugly fits a whole cut out of it! I discuss ths shunyasutra duality elsewhere . Here Normal usefully calls it a tracheotomy. Nothing is in fact something in our perceptions and it is the absence of comparable differences in perceived form . The forms I meditate on are quarter arcs and spheroidal octants Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 08, 2017, 11:48:58 PM http://m.youtube.com/watch?v=6XghF70fqkY
Here Norman introduces the bi vector and trivet or so called in linear algebra. The product of 2 line segments was and is an idea that requires explanation. The basic explanation is design. A product, unlike addition and subtraction is a designed notation or label. Addition and subtraction are natural space like dynamics. We can consider aggregation and did aggregation for example. But multiplication we do not see dynamically or naturally. What we do see is duplication or copying an identity, sometimes not exactly identically. So we see multiplicatr for es. Almost the same if not exactly the same, produced by duplication in sace- like ways. Logically this was untenable, one can not prove identicality, it is agreed or assented to. To build a logically sound system Grassmann took indisputable dy amicus and then based everything on those dynamics by simply analogising the dynamic carefully. It is explained in abstract detail in the induction, but in the first chapter it is set out as an inductive synthesis! Thus the first step is set and the following steps are synthesised by a process that inductively uses the prior step to get the current step. Then the process is applied again to the current step to get the next step. And so it proceeds. One of the consequences of this was that if a line a labels the points between A and B, the beginning and ending points of a dynamic thenm-a was set to lable the dynamic from B to A . The inductive method for consistency meant that ab the continuous form that starts a line a and finishes at line a' by parallel transport in the direction of line b. Notationally then and inductively Hermanns insisted we start with a and finish with b thusba not only had to be the the reverse of the previous synthesis process it also had to take the negative sign when corresponded or equated to the geometrical form. What made it imperative was the strict application of the inductive process itself. There are definitional ways to condpstruct an algebra and Norman and others do this freely, but as far as Hermanns was concerned this was too arbitrary, and not logically defensible, whereas no one can impeach an inductive synthesis, except to say how weird it is to the common sense! But it is precisely the common sense that fails when one ventures into higher and higher inductive steps or stages or dimensions. The inductive process gives certainty by giving invariant process no matter what the outcome. It is also an amazing discipline of construction or synthesis, so much becomes consistently manageable , consistently structured and confidently applicable. Hermanns develops the theory by using elements and confirms the solidity of the labelling pattern. However he designs different products as he develops his expertise . The spreading apart product, which is based on inductively designed is accompanied by the colliding together product which is designed to use the perpendicular projection of the right triangle . Then he designs a quotient product which enables him to do rational and inverse computations algebraically, and thus represent division as analytically finding the factors in a multiplication process. There are other products he designs some not as useful as others , and by 1877 he was able to demonstrate how the methods he developed designed the Quaternion algebra. The doctrine he espoused was a system of expert analogies and intuitions based on rigorous research in combinatorial processes that have their basis in natural dynamics . The essential creative entity was and is the dynamic creative point that creates the line, whether curved or straight. The Erlangen project of Felix Klein, the transformational Geometry he espoused is in man ways a take on the apprehension of dynamics required for the modern world, the Leibnizian analysis of the way forward for Natural philosophy and physics and a part of the appreciation of the failure of the mathematics or geometry of the time. The reforms in Prussia led to a Prussian Renaissance that Gauss tried to steer through his Protogé Riemann, but in fact he ignored the Genius of the Grassmanns who were nearer to the solution than any high academician Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on March 11, 2017, 08:47:45 AM http://m.youtube.com/watch?v=rz8A5l_yn34
The coining of the ideas of covariance and contravariance take me right back tomHermann Grassmann via Peano. The contravariant idea is what Hermnn called the Stepping apart or spreading out product, and of course the covariant product is the colliding together product . The covariant remarks upon the coming together of the variation, typically through vertical projection or a dropping of a perpendicular. This is such a general rule that one hardly bothers to find out where it comes from. Certainly in Eudoxus and Euclids time the quarter turn was more useful than a general perpendicular Except where computation was required, then Pythagoras theorem was indisputably necessary and the adjacent limbs of a right triangle were thought of together , ans a varying together. The actual calculated variation took centuries to tabulate to a useful degree of accuracy. These measures are as valid in spaceship navigation as in seafaring navigation! The contravariant behaviour involves the parallelogram. As many know cyclic polygons exist within circular boundaries, but mus vary covariant ly, that is as the right triangle varies, but once a constraint regarding the diameter of. Parallelogram is applied it becomes clear that the sides have to vary in opposite ways . The angles at the diameter increase while the angles at the other non fixed diameter de tease. . That diameter extends while the other reminds constant . The constraint of parallelism ensures this contravariance in the angles. Of the vertices of the parallelogram. The astute observer will not that the the vertices of a right triangle also contravary, but on suchnacwaybthatbthe non diameter angle is always 90°, thus theybcoopperatebto implement one another. One another, to complete on another in that sense. In the video you see how a constant perimeter parlllogram rotates within the constant trochoidal,dynamic , but as the perimeter busies so does the size of the Trochoid. . Thus contravariant and covariant measures serve to describe trochoidal dynamicsv Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on May 30, 2017, 12:34:17 PM http://m.youtube.com/watch?v=e17J0mOhelQ
In making this presentation Norman once again reveals the power of The Grassmanns et. al. Work on lineal algebra. The dot product, the cross product, the matrix form and the determinant are all carefully placed to unify the rational trigonometry he developed and highlights. However, much of this work is based on traditional mathematical ideas at primary and secondary level including the derivation of the Cosine formula. What is sidestepped here is the use of or need for trig tables because he is not interested in the arc "length" or rather arc segment associated in these tables. It has taken me a while to apprehend where my maths education was deficient and actually mislead me from the use and meaning of the trig tables, which were developed for astrologers who sought to place stars on various spheres in the night sky in such a way as to match their observations. The tables associate arc length to Pythagoras theorem or in terms of forms the arc segments arc sectors, chord segments etc to Thales theorem of the right triangle in a semi circle, through the ratio of squares on the right triangles in a given standard circle. Reducing these line and arc segments to lengths creates a problem of measurement, whereas considering them as dynamic extensions means that a rolling disc can connect the arc extension to the diameter of any given disc. Thus essentially every formulation for a straight line segment can be factored by Pi to apply to an arc extension. The dot product and cross product of an arc segment with a general other arc segment or the cross product of a matrix of arc segments can thus be computationally and algebraically derived, without the use of tables. Tables become useful when a specific circle or sphere is given so that actual approximate measures can be computed. Thus for example given a wheel of a given diameter tables enable one to calculate the movement of the centre along the diametr for a given radian arc length. In fact the radian arc length is designed to make that calculation particularly straight forward. Given a fixed centre one can measure the circular velocity along a given arc segment given the radius and the standard radian measures. The general spiral motion therefore is a dynamic combination of circular arc vectors and line vectors, and it is necessarily approached by bitwise dynamic approximations familiar in the ideas of calculus, but readily generated by a fractal generator! It is this apprehension of the dynamic application of forms in a fractal generator by means of the inductive steps that liberates the presentation of calculus as a scripted set of notations into a dynamic accessible plaything which does so much more than calculate numbers either as areas or rates of change, but through the distance estimation, the surface plotting, the colour cycling the constraints produces in a modern computer a viewable form, The issue is to take that process and understand the product and see if it gives insight into our utilisation of natural processes in technology and apprehending those natural processes in a rational way. And yet we must avoid the conceit that our meddling models are in anyway fundamental laws of our universe, and humbly confine ourselves to developing some expertise that may have some practical use in our society. The reliance on the mathesis of the imaginaries has no place in a modern apprehension of measure and calculation . It is firmly and definitely a measure of circular arc extension starting with the natural semi perimeter of a circle but more usually based on the quarter arc, both related by the right angle they contain. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on July 26, 2017, 11:52:54 AM Circle theorems based on " angles" are really Based on Arcs
http://m.youtube.com/watch?v=opXl43e254Q Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on August 29, 2017, 11:19:50 AM I am always fascinated by the problem of trisecting a general angle. The reason being when I was a child I was introduced to engineering drawing one cold autumn day, and shown how to draw a sine curve from a circle .
I grasped it fom a co ordinate perspective but not from a dynamic one. How could you draw. Circle moving ? The ideas are straightforward, but I had obscured important steps from my minds eye for so long it seemed n impossible task, I knew it was not but I could not remember why. These videos helped me remember , and also suggested another approximate method to trisecting the general angle, by using a cycloid! Circle theorems and constructions are fundamental to trochoidal dynamics and kinematics . https://youtu.be/EHMZkYhOX0E Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on September 06, 2017, 09:46:27 AM http://youtu.be/L24GzTaOll0
http://m.youtube.com/watch?v=L24GzTaOll0 This has caused a bit of a sensation in the news, but it is August when news is traditionally a bit less seriously reported. It's worth understanding that the Arithmoi were regular geometrical forms not numerals. So a measure was counted , and different measures were utilised to count. . For construction areal measures were utilised for accuracy and length for example was not considered apart from a standard figure. In this case the Square was the standard and lengths and areas and corners measured by it in various ways. The measurement of tri corners or trigonometry was done by ratios of squares . The squares were divided into sub squares of the standard square using 60 or some power of 60 60 was chosen because of its factor tabulation. Factors 1,2,3,4 ,5,6,10 are immediately available. Using 120(2x60) , 180(3x60) , adds factors 8 , then 9. 420( 7x60) adds factor 7. So all the factors 1 through 10 are available through on multiple of 60 Exact calculations are therefore discoverable for many fractional or sub unit measure. We have been taught to normalise our tri corner ratios on a fixed radius, they normalised on a fixed square side up to a maximal diagonal.they avoided that ratio precisely because it was irrational. But they went on to calculate approximations for that ratio to a useful level of accuracy. The exactness of there counts depended on what multiple of 60 they used. For example 2x3x7x11x13x60 gives them all the factors up to 16 . You will note the " prime " factors are what extend utility of the base 60 factorisation tables. Within these tables suare counts and cube counts can be identified. The factors 3,4,5 are significant because the Pythagorean triple can be expressed as (4-1),4,(4+1) and that pattern is found in their general process of finding squared diagonals. Proposition 1:14 explains the Egyptian Greek form of this circular relationship. Thus we see the Arithmoi a pattern of forms represented by a pattern of pressed orms enabling the tables of Gometry. Pythagoras is said to have pointed this out: without the Arithmoi no one can perform geometry! The factors of 60 involved in this triple enable us to find 45,60,75 and 60,80,100 very readily. The 5,12,17 triples require us to extend the prime factors that multiply 60 to include 17, we can hunt for triples using this kind of process. Prime numbers also rapidly become apparent, and their use as defining new factorisation tables evident. Thus the name proto or first in the Greek. They are the first Arithmos of a new factorisation paradigm. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on September 10, 2017, 08:47:02 AM http://youtu.be/I9a2KK3RqUw
http://m.youtube.com/watch?v=I9a2KK3RqUw Norman ties down orientation to cyclical rotation. He implies that lineal orientation is subordinate to rotational orientation. Thus Within a a cyclically ordered triangle the ordered side AB has to be set = to -BA , and within an ordered parallelogram BC //DA means DA=-AD=-the vector BC. The order links the ordered side to a parallel vector/ Träger or line segment on that construction line and to its reverse or opposing vector by mirror reflection or half circle rotation. What about quarter rotation? We find the signed "area"/form under cyclical rotation behaves exactly the same as i the magnitude for rotation by a quarter turn . Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: Alef on September 13, 2017, 03:24:04 PM It's suprising that one of the most important ancient greek term, the logos is largely non understandable in modern west. Jet analogos, biologos ...
I writed a few things about neoplatonics. mystics and history of mathemathics and this was very usefull. Al Khwarzimi in arabic means from Khwarezm what is to Bagdad as Bagdad is to Rome. Maybe for Europe they would look as some strange and very underdeveloped islamic russians. But he bought indian mathematical knowledge to the Bagdad what become arabic numerals. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: Alef on September 13, 2017, 03:47:22 PM Quote Saptabhangi - jain seven valued logics. 1. From one perspective it is. 2. From one perspective it is not. 3. From one perspective it is indescribable. 4. From one perspective it is, and from one perspective it is not. 5. From one perspective it is, and from one perspective it is indescribable. 6. From one perspective it is not, and from one perspective it is indescribable. 7. From one perspective it is, from one perspective it is not, and from one perspective it is indescribable. Thus logical: true - t, false - f, indescribable - i. So seven truth values: t, f, i, tf, ti, fi, tfi. One perspective here is one point of wiev. Indescribable here is that we can't be sure of the adequacy of desciption of what there is. Some wanted to implement this in IT as "eastern logics". It's like Ghandi that truth looks different from different perspectives. But realy it is how they in India can coexist together. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 14, 2017, 10:31:23 AM Thank you Alef, not only for contributing but for putting up with my many typos.
The Pythagoreans were influence by all the great civilisations, even though we know very litte about Pythagoras, at present. But it is strongly suggested by the PIE language studies that Indian or Dravidian influences were felt worldwide including Sumer, and Egypt and the Americas. And of course the Chinese . Greece as a coastal nation thrived on worldwide trading links beyond the Mediterranean ! The 7 valued logic shows how the west darkened it's eyes to the wisdom of the ancient sages! Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 17, 2017, 05:11:20 AM The Stoikeia begins;
A mark is not a part of anything. And thus a drawn mark is an extent without form And also a drawn mark is bounded by marks. And thus a good drawn mark is one out of which dual marks lie upon itself And thusly a surface upon which light shines is a unit coming to be of extension and form . Thus a surface upon which light shines is bounded by drawn marks. A footabl surface upon which light shines is one out of dual marks of which the good drawn marks lie upon itself. And so a footable surface upon which light shines knee bend is where in foot pacing two drawn marks streching away from one another and not upon a good drawn mark placed together , towards one another the drawn marks meet . The descriptions are dynamic and active. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: Alef on October 17, 2017, 04:21:07 PM Eastern influence could be less know becouse western science just don't know mutch about eastern nations. Ancient persians tought that there exists different spiritual and material worlds what could influence Platon's toughts on world of eidos of ideal objects. Hussar and khazar could be form persian for 1000 - military unit.
Origin of basics of mathematics and even our culture is not so clear. Indians could take some ideas from west but west could had taken ideas from India. Shining nimbus around the head could had originated in India: (https://images-eu.ssl-images-amazon.com/images/I/61aVUzs%2BrML._SY300_QL70_.jpg) During ancient world there were no sutch divide of civilisations and east and west - realy a religious divide. In chinese culture they use a lot idea about elements what in west was popular in greco - roman times. So I made entry about Al Khwarezmi, algorithms and numbers: http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20362 (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=20362) But numbers and 0 is indian. They just came here by arabs. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 18, 2017, 04:24:08 AM As you continue your meditations Alef may your insight prove rewarding and useful. . For the form seen with the eye( eidos) is distinct from the form formed by the hand with clay( plassos) . And Socrates played a game with men of intellect only, and so did Plato.
The concept of being, or ontology for a man imprisoned and shackled is distinct from that which he may form as a free agent. In a society, culture or family who really is free? Thus the sages of the east and far east despised no one of humble birth for they saw that greatness was reincarnated and had to be sought out by the wise. In the west some bad men sought to extinguish this light in greed and madness. Those who treasured the eidos were gifted to form the plassos by the Musai. Or those who formed the plassos perfected it in the eidos and treasured their own creation, what came to be through their activities. . Do you accept fact or myth? The wise say accept both as illusions and enjoy your life. Thus fractals generate joy for the wise and terror for the uninitiated . Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: Alef on October 18, 2017, 04:36:57 PM At these they didn't had democracy as we understand it.
But maybe divide of east and west started with the medieval ages and religious divide. East, west and the south could be this: (http://www.hf.uio.no/ikos/english/research/network/ikos-austin-collaboration/Graduate-Conference/map-jerusalem.jpg) Suprisingly greeks believed in reincarnation and pythagoreans was one of them. The recognised that some of the knowledge they learned in Egypt or other more exotic eastern countries. Title: Re: Principles of The Logos Kairos Sumbola Sunthemata Summetria Theurgigical set Post by: jehovajah on October 19, 2017, 01:35:43 PM http://youtu.be/EiP3Jf5hnAI http://m.youtube.com/watch?v=EiP3Jf5hnAI Starting with lecture 37 Norman presents his Förderung on trigonometry . Essentially he eliminated the need for tables of trigonometric values. While Norman appears to not respect circular measures, this is not the case. Circular functions are normally truncated infinite series. The question is : how useful is this computational load? How practical and how accurate? While the Pythagoreans had no fear of incommensurable magnitudes, nor of approximations they nevertheless highly valued precise or exact results to processes. They preferred to contain approximations between 2 exact results wherever possible. In book5 Eudoxus opines that magnitudes in a ratio or logos, meaning a situation of study about which many words may be said or written in comparison or contrast, when honed down to extensive magnitude has 3 cases. The magnitudes are dual or equal to each other by duplication or fitting precisely upon each other or some other process of comparison or production. If they are not dual then the cases are , in order, the first is greater than, or the first is lesser than the second. The study of these cases are not limited to static extensions either, for then the dynamic reality in which we live wold be unrelated to our studies, and in fact nothing could be developed, dualed or multiplied! In his study of magnitudes Eudoxus starts with the drawn mark. A continuously moving extension. Thus continuity is inherent by motion. What creates this mark is an inherent power within either the mark or the observer/ creator / producer or both. Thus a drawn mark may be marked in segments or parts at will. Lesser drawn marks are parts of greater ones, and greater drawn marks may be apprehended as multiples of lesser ones. It is the relationships between these partitions of wholes or dualing into multiple forms of wholes that is explored in book5. The most used one is the Logos Analogos relationship. Here Norman makes use of this basic relationship to depict spatial extensive magnitudes of interest. But he uses advanced ideas from book 7 to establish his laws. In fact the root of these advanced ideas are in Book 2 of the Stoikeia , where dynamic parallelograms and rotations are normalised by the quarter turn or ortho. In nook 2 the metrons established in book 1: trilateral quadrilaterals, trigonia and their relationships are used to establish quantification or metrication. We can define counting, and a counter or scalar magnitude in book 2 . That counter is usually a parallelogram or a dual drawn mark. A normalised parallelogram is called a Rectilineal parallelogram. There are corners in these works but no degree or radian measure. The measure that is used is the chord of a circular arc, and this is compared to the diameter of the circle in which it is formed. The quadrance and the spread thus relate to squares drawn on these chords and diameters. However, the spread is specific to lines emanating through the centre of a citcle . The spread relates lines intersecting marks with that circle to the ratio between the square on the perpendicular and the square on the radius, that is it uses the quadrances of these compared line segments in a circle to define spread. The chord could have been used but we are used to half angles( corners at the intersection of 2 lines bisected) and these help in 3 dimensions considerably, especially in cases of rotation. Why did we spend so long calculating tables? The Sumerians set the trend, and of course they had no uniform view of a normalised reference frame. Despite astrologers fixing the ecliptic and many other quantity standards , the use of tables conducted the mi d away from establishing a normalised reference frame in the spheres of reality. Or rather, the absence of the idea of a vector /Träger did not stand out as a simplification of calculation processes, or a way to present calculations algebraically in relation to extensive magnitudes. We owe many people a debt for this insight but foremost must be Justus Grassmann who reduced his eras learnings to these simple fundaments called vectors/ Trägeren. This inspired Robert and principally Hermann to publicise these ideas, even in the teeth of Gauss and Riemann ! Sir William Rwan Hamilton deserves mention, but it is the primary educators the Grassmanns who rolled it out where it could grow! In the minds of young students. http://youtu.be/xiQNkF_svVw http://m.youtube.com/watch?v=xiQNkF_svVw The issue with sin^2 is the indeterminacy of acute and obtuse corners, which is why the chord is clearer. However, the geometry has to be consulted to determine obtuse or acute reference. |