Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

Commentary

There is so much prior work referenced that it is clear that the manipulations and transformations Hermann makes from the product design specification to the product design implementation must appear slightly mysterious. But a good exposition of Hermanns thinking is explained in Normns famous Maths problem 13 series.

Here the outwardly completing entities are line segments normal( now meaning perpendicular) to the plane of the other 2 . This is important because Hermann reduces his notation by this device, but also rotation in the plane and space is shown to be notated cyclically!

It is this cyclical notated rotation that took me so long to uncover, and its importance is that the cyclical rotation in the plane determines which normal line segment is chosen!

In the set up this is so straightforward that I marvel at how difficult it was to perceive!

Thus e₁e₂ must pick e₃ as the norm ,e₂e₃ must pick e₁ as the norm, and do on. And of course if you reverse the cyclical order you must pick the negative norm.

While it is very true that cyclical rotation does not require a norm, or an axis of rotation, it is also true that in any normal system there is an intercommunicant norm specified by this outwardly completing process. The point is that it does not express the specific axis of rotation for that plane. The rotation is specified by the perimetral cycle!

In addition the norm as a perpendicular associate is  transmutable into any general associate  third element line segment  in a 3 independent planar element system. The rotation is still provided by the cycling round the perimeter in the plane.

[jehovajah]

I have now completed the notation in page 378.

The ergänzung process provides an alternative notation , but of itself is not a rotation.

I have come across this alternative representation in my own meanderings and wondered how it could ever be useful! But, using the concept of a parallelogram provides 2 possible utilisations: the first is the diagonals, the second is a circular arc or swinging arm within the parallelogram. Neither of these have been empirically tested or founded, so it is of interest to see how Hermann proceeds with his development and design, and what constraints the law of associativity places on the possibilities.

My concurrent work in the first chapter of the Ausdehnungslehre has thrown up the innovative manner in which Hermann couches the development

[jehovajah]

Yet another misconception begins to fall away! Because I have been brought up on axes and line segments as determining factors, and initial guides I failed to appreciate that a flat figure can be a determiner, the so called bivector ,

If I draw a line it is visible because it has width and de,oth of tone related to depth of material deposited or excavated on or in the plane.  Thus an oriented plane figure should be considered as any line segment a directable form, a project able form . Thus we can add and product flat figures similarly to line segments..

So returning to the outwardly completing entity it suddenly dawned that just as an outwardly completing entity to a flat figure is a line segment determining the third element of the spatial " reference system , and subsequently an outwardly completing entity for a line segment is a plane. Thus for a whimsical line segment the outwardly completing entity will be some whimsical plane to which it is perpendicular, or which is  corresponding to the relationship between the primitive elements.

Thus, rather than trying to find vector or line segment ensemble that forms a reference frame I should be looking for a plane or surface through which this whimsical line segment passes " perpendicularly".

[jehovajah]

The Place of Hamiltons Quaternions in the Doctrine of Extending/ Extensive Magnitude

By  H. Grassmann.   In Stettin
--------------------------------
Page 378
These monad products let themselves guide back concording to the formula
ab  = [a | b ] + |[ab ] onto the colliding in and spreading out multiplication. For both these are, concording to the above monad products,  tied up alongside the formulas


Where r and s are two differing entities of the Indices 1,2,3.

Thereto still to come the everyway pleasing of the above label of the outwardly completing  entity, the formulas

If r,s,t  are related to the "cyclic group" -cyclically interchanging- 1,2,3; that brands, r,s,t either  = 1,2,3   Or 2,3,1. Or 3,1,2

Hereout follows for the Middler multiplication of the monads   the constraining rules


If r,s,t  are related to the Cyclic Interchange group 1,2,3.

Then output results itself for the Middler Multiplication of three monads, if one the cyclic assigning from r,s,t holds fast

Plainly thus


That brands, for 3 differing  monad factors everyway unioning ability empowers. Plainly thus for 3 like entities.

Thus also for two like entities which are separated through an unlike entity .

Because
There  against exists and
Therefore also for this case everyway unioning ability Should empower, thus must necessarily exist

Turned around: if exists, thusly output results itself also for the yet remaining products out of 3 monads everyway unioning ability  of the factors.

 Because then is ; further

and

Then it follows, therefore: the everyway unioning ability for every 3 monad factors, therefore for every 3 factors, therefore also for whimsically many(&1§3.).

We set therehere for the Middler multiplication , while was set, therefore
ab  = –[a | b ] + |[ab ]

Out of this fundamental likening  follows all the rules of the Quaternions, and indeed certainly all with the greatest of ease! Also the Nature measured out benaming output results itself here concording from self.

We can name to be: –[a | b ] the  colliding in part and |[ab ] the spreading out part of the quaternion.

Be a and b parallel, thus comes to be the spreading out part null, and the factors how by considering each colliding in product everyway toutable ( for exchange) come to be .

Become a and b to one another perpendicular/ vertical, thus comes to be the colliding in part null and the factors how By considering each spreading out product with sign change everyway toutable ( for exchange) come to be

Everyway touts ( for exchange) one the factors of a Middler product, thus remains the colliding in part un-everyway-varied, the spreading out part varies its sign(—/+)

Also I will  besign, upon the succeeding part: the tally marks continuously with Greek and the line segments continuously with Latin printers block typeface. Only the printers block q  for the besigning of the quaternions I will protect, upon ( the following part) .

[jehovajah]

Commentary on page 378

What is a quaternion?

To answer that question I have to ask what is a number!

Here I have resolutely put Tally mark to express the nature and role of a numeral. But the notion of a " number" derives from Arithmos, and that is a mosaic form or rather thought pattern perceiving "spatial objects", let say.

Spatial objects are thus magnitudes which in comparison may be quantified one comparator by the other( usually the greater by the lesser). So these mosaicised objects we manhandle in space , rotating, translating them this way and that. But our notation, the marks on the page necessarily hold everything still.

When doing " geometry" the difference between those that get it and those that do not is the ability to rotate the written page , translate it, scale it etc physically and thn ultimately mentally.  A still diagram on a fixed page is Mathematik to Astrology and thus Geometry. These are dynamic fields of study!

So formally the work of Bombelli in particular established the concept of a dynamic plane rotating and translating and dilating notation called a complex number.

Well originally it was called an adjugate number by Bombelli, but later it was called a complex number because of its complex form and role.

Hamilton wanted to know if a three dimensional complex number existed, or could be constructed.bhe felt that it would be the key to describing 3 dimnsional rotations by calculations. Rodrigues tackled the issue of rotation astronomically nd thus geometrically in space.

The problem was not hard, it was just long winded and tedious and required many pages of set up. The Electrodynamic theory , or the equations of Relativity suffer from the same difficulty. It is so easy to get lost in the symbols, diagrams, stages , subsidiary results etc.

The complex numbers had revealed an elegant and accurate method for calculating the angular displacements or positions in a rotation. Hamilton wanted that kind of simple beauty for rotations in 3d. Rodrigues simply wanted to lay out the important parts of the calculation even if it was a long winded way to do it. It may have been the only way to do it.

One must not get the impression that no one knew how to describe a 3d rotation, because that is not the case. This was known from Ancint times and in particular the Conics were the solution par excellence. The issue simply was that it took days of calculation to get a result! All but the intellectual giants like Gauss, LaPlace, Euler, LaGrange found these calculations intimidating and tedious. Errors eventually crept in and becme a problem of their own!

In fact Gauss and Boltzmann strived to find ways to minimise the effects of these mistakes in calculation, eventually giving rise to statistical and probabilistic Mechanics .

Was there a more direct calculative approach like the complex numbers? Was there a more intuitive pproach like astrology and Geometry? Leibniz thought there was, but he did not know how to get at it. Non of his Europen ollegues or correspondents were interested in finding it anyhow!

So we come to the Grasmanns. They took the time, father and sons, to review the Whole of Mathematics and logic  from the ground up. They were part of a small international group of pioneers of Ring theory, which included Abel and some crystallographers. They were convinced Nature has a better art or skill for measuring And "calculating " natural magnitudes like crystal spacings and dynamic motions say of ballistic entities. By looking over everything philosophers had achieved so far they hoped to find and did find more general and more streamlined processes.. But they foundered on Multiplication.


It is Hermanns rigorous investigation of Multiplication which I have called product design, after a suggestion by Hermann, that was the key. It revealed that we did not understand " Multiplication" and how it is contextualised nd constructed. In particular it revealed many bad nd misleading practices by Mathematicins of his day, especially Prussian ones!

We have to start with space, the astrology and geometry of it, how we respond to its dynmic forms nd structure with our on internal thought patterns. These thought patterns we construct as representations of the I pact of these Dynamics on us, on our thought processes.

We then have a duty to rigorously test and manipulate these thought patterns to find out if they have invariant properties despite always changing!

As Hermann pointed out, if we do not look for invariance and find it then we are just going to be in and end up in a mess!

It is these invariants that enable us to generate rules and constraints on applicbility. These rules and constraints we write in a rhetorical shorthand called a label or symbol. These rhetorics express notions, concepts ideas, and the symbols are meant to anchor these to remind and recall them at will.

However, we soon find out that the symbols carry a precious advantage of their own: the same structural arrangment of symbols can represent many differing contexts! The symbols themselves have an invariant structural property.

This being the case the constraints take on an important role in the interpretation of these symbolic structures. They help define the applicable context.

Structural symbol collections of this nature are often called Formulae, Expressions,Identities, even Laws of Nature! But they are essentially empty or " abstracted" thought patterns which applied by Analogy across many fields of study acquire a powerful organising myth and a mythology.

Some of these myths we give names too, like the complex numbers. But some of these myths are named by their original constructors or discoverers like the Quaternions.

This powerful group of 4 , like an elite group in the regular army has a special connection to the Pythagorean philosophy and Mythology.

So the myth of Quatenions is based on a collection of constraining equations and interrelated products formed around the right triangle . This right triangle is not one but 3 in different planes in space. These constraints and products can be manipulated by specific methods of calculation to give the angles and lengths of a dilating rotation. In particular a particular set of values for the constraints enable a pure rotation to be calculated by the products, that is the angle or the orientation of a line segment in space.

When Hamilton finally arrived at these products and constraints he  did so after a welter of other investigations . His initial presentation was thus overly complicated and difficult to perceive, but his reputation meant that many leading researchers adopted this termiginous presentation without grasping its proper use. Many mistakes of fact followed, and mistakes of sign carrying! Those intuitive geometers and physicists like Lord Kelvin hated it. They could not " see" what was happening, and preferred Rodrigues approach.

However a few had found another approach by an unknown " Professor" Herman Grassmann. When Hamilton read the 1844 Ausdehnungslhre in around 1853 he immediately rewrote the Quaternions( that is he revised his approach) to present them in terms of Line Segments rather than Coordinate Geometry. But by then Gibbs had stolen the hearts of his intended Audience with his baudlerized versions of Grassmanns ideas. Vectors stole the easiest ideas from the 2 approaches and conquered the hearts of the new Physics community.

Later Bill Clifford did the proper analysis of the two approaches, and placed quaternions within Hermanns general structures. The dot producs design and the cross product design were emphasised and so the vector algebra translations began, forgetting the foundational work of Hermann as too hard.

Followers of Clifford rebranded Grassmanns work as Clifford Algebra and developed their own Nomenclature for his ideas.

What we are reading on this page378 is Hermanns product design and constraints for the method of calculating rotation dilationl. A quaternion is an extending magnitude that is manipulated by these product designs and constraints. They exist only in 3 dimensional (or less ) systems, and they require 2 arbitrary line segments to define them, within a rectangular Cartesian coordinate reference frame( although they are defined in any 3 dimensional general coordinate system)

At this stage Hermann has reduced everything to normal "reference frames", which in his set up are 3 mutually orthogonal line segments, but later he will show how it extends to any 3 independent line segments( or rather he has demonstrated this in previous papers referenced in the footnotes if not in the body of the text).

Although you may, like me, wonder how an equation can equal a quaternion, but you must here remove the word equation from your vocabulary. Hermann uses likenings or analogous expressions! Thus a product of line segments gives a quaternion expression as a result. Later he will show how a product of quaternion expressions  gives a quaternion expression as a result.

These results are methods. The methods show how to quickly calculate the colliding in product and the associated spreading out product. The results of these calculations contain the orientations and length of the rotation dilation.

There is also a lot of geometrical intuition contained in this presentation of the product, so one can quickly image or imagine what is happening dynamically by the calculation.

I shall discuss Hermanns method and the format of the Quaternion or Middler product and the role of the cyclic interchange group after I have done the Latex editing. It is clear that Quads are not obvious in the fundamental likening.

[jehovajah]

Commentary on page 376-378

I have recently translated chapter 1 §§ 13-14 of the Ausdehnungslehre. As usual it is surprising!
The content of what one is doing , the specifics one is trying to communicate are deliberately absent. The Structure of what one is attempting to do, or build is the expertise Hermann is conveying, teaching or rather more hypnotically inducing!


The rich content of the 1844 version serves many purposes, but it's direct goal is to promote a certain state of mind or mindset. This is the Förderung Hermann immediatly mentions. In fact later on Hetmann mentions a second mindset: the Hegelian philosophical and inspirational one, based on a meditative homiletics Praxis!

Those who have studied Descartes writings will know how important his daily Praxis was to him and his creativity! In fact it was ue to that praxis being broken by command of the queen of Sweden(?) who insisted he get up and teach her at very early hours on freezing cold mornings that he " caught a cold" of which he supposedly died. Certain habits and practices once installed are not to be messed with!

I would advise the meditation on the Ausdehnungslehre as a recommended Daily Praxis for any philosopher of mathematics, physics and computational sciences, and indeed psychology of creativity and problem solving.

So, because I value Nomans insights into mathematics I have been mainly guided by his understandings of the foundational aspects of mathematics, along with my other guides, but now I have to place Hermann in a pole position . Now I have to review all my meanderings in the light of a better grasp on what Hemann is doing and How he is proceeding to carry it out.

So the background to how Hermann designs and establishes or instantiates the Middler product I have no doubt will reveal more surprises, but here we simply have a masterly set up to the labels required to "solve" the problem of how to find an Algebraic representation of rotation like the so called complex or imaginary " numbers". In fact Hamilton thought of the whole thing as a Mathesis, a Mathematical doctrine that ordained the behaviour of its content!

So here we see the product design, the elementary products used to instantiate the design, the elements that are products of prior creating processes , the constraints that they have to fulfill and how that constrains how we can think about the pattern of their behaviours. We are also, iin passing introduced to invariant transformations, and we are immersed in all of this in the consisten background of 3. Hernann and Hegel consistently and intuitively as well as deliberately designed around 3, at the level of 3 and at the output result of 3"

The use of the Associativity law/ rule/ design parameter is genius, page 378 is all about the full demonstration( up o background referencing) of that constraint and how it structures the output results and behaviours of the colliding in, the spreading out and the Midfler products in terms of the Factors.

The design brief is to ensure abc =a(bc)

When I first started to design a fractal equation for the 3d Mandelbrot push, I was much helped by David Makins use of product tables .

I had no prior knowledge of them, just the basics of what David set out and a aural but wrong belief that they were multiplication tables . It was only later that I recalled Modulo arithmetic, and Kujonais work on poly signs struck a chord with my meditation on unary operators . But frankly I was just blueskying it, flying by the seat of my pants. That is when I started the polynomial rotations thread. I did want to start a matrix multiplication thread, a subject I knew hardly anything about! But a man can only do so much, and I was deep into Fractal Foundations of Mathematics and the sets FS and notFS at the time.

At the time I hated group and ring theory . My introduction o it at university was a negative experince! I skipped all the Tutorials or worked on my Anlaysis papers during them if I recall correctly.

So Modulo arithmetic based on Eulers circles were about all I could cope with.

I have to thank Norman for his introduction to group theory course for turning that round! It was while going through that course that I designed Newtons Triples. And thanks to Alef and others rendering my product design I caught a glimpse of how group/ring theory was at the heart of generating the Mandelbrot set .

So why did I turn to translating Hermann?
Since working very heavily on Hamiltons Quaternions nd corresponding with Doug Sweetster, the stand up Physicist, I had heard about Clifford Algebras and Grasmann, but I thought that Hamilton was " the real deal", and Doug and I clung on to Quaternions as some mythical," solution" to everything.

However I am a researcher and I researched Grasdmann briefly , found a free downloadable manual by J Brown on Grassman algebra for Mathmatica, and googled Grassmann on YouTube and Found Norman Wildberger. So I felt I owed it to myself to sort this mess out and find out why Grassmann was hailed as an unsung folk hero and geniu!

So here I am today confirming that, and also my initial impression of Jiggery pokery, sleight of hand and general fooling the public as being correct! But it is not Grassmann that was doing this , it was everyone else!

In this page 378 Grassmann summarises in his notation all the rules I sweated daily and nightly to elaborate in tables in the polynomilal rotations thread and in the V9 thread. And he does it imply by design!

The outwardly completing entity is not a product, but an identity, and yet he has introduced it into a product design! When we look at the demonstration of the associativity rules it " behaves" and feels and looks like a product equation string. But in fact we have to stay in the mindset Hermann has laboured to induce in us. These are not equations but strings of like things.

What these strings of like things do is enable us as the thinkers observers or constructors to guide our minds around or back to different " points of viewe" or "statements/ expressions" of general and particular arrangements of elements. It is us the observer who are changing our relative status to the content of the labels.. And we are doing it in an agreed and reasonable manner before we get to any specific details!

The cycles or cyclic interchange group/ ring is a fundamental " dynamic" device. It controls the transformation of 3 elements In precisely the same way that the outwardly completing entity behaves, and thus automatically establishes the rotation of our relative positions .  Because of this Hetmann is able to drop some notational markers in explaining how it operates associativly

The final Middler product relies on the colliding in product and the outward completing entity of the spreading out product! Remember that !

The cycles controls not just rotation in space therefore, it controls, through the outwardly completing entity definition the mental orientations of the observer to the notation. The – sign therefore is a fundamental indicator of mental orientation, much more than it is of spatial orientation. Spatial orientation is just one interpretive instance of the – sign

All rotation or sign change is based on this cyclic interchange group for its behaviour in Hermanns definitions of rotational transformations. We do not need to use "imaginary numbers" , because the modulo arithmetics based on Eulers circle or clock arithmetics are precisely homologous rings.

[jehovajah]

You may wish to watch this video to see if any " familiar" notation appears and in what context.

<a href="http://www.youtube.com/v/Xt1zAWbccIQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Xt1zAWbccIQ&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/Xt1zAWbccIQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Xt1zAWbccIQ&rel=1&fs=1&hd=1</a>

Remembering that Hermann is only seeking to demonstrate that the method of the Ausdehnungslehre is more than equal to the task of expositing quaternions and more especially giving them a more accessible geometrical meaning and relation

[jehovajah]

I was going to put this in the Ausdehnungslehre  thread, but I decided to put it here to show how the Middler product is made up from the colliding in product and the spreading out product of of 2 general line segments, transformed by the outwardly completing entity.

While the author brilliantly evokes the Abschatten product which is Hermanns original concept of the colliding in product, he then goes combinatorial for the spreading out product. He does not know of the outwardly completing entity, but uses Gibbs idea of an area vector. This is precisely where Gibbs lost understanding of Hermanns set up.

The combinatorial aspect is explained in Ausdehnungdlehre chapter 1 §13. It is an identity transformation through labelling. The thing labelled is the spreading out products. So each of the systems of elements( second , third step) laid out are label versions of the spreading out products . The geometrical objects are line segments , plane segments and block segments ! To come to thst view you have to use the outward completing entity . This is why the outer product is designed with the outward completing entity symbol.
<a href="http://www.youtube.com/v/f5liqUk0ZTw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/f5liqUk0ZTw&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/f5liqUk0ZTw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/f5liqUk0ZTw&rel=1&fs=1&hd=1</a>

[jehovajah]

Consider the group 1,2,3. Some may use set notation or even combinatorial notation, but such distinguishes are not necessary at this level. And by the way these are numerals or symbols not numbers.

This is the simplest group in which I can show cyclic interchange. The group 1,2 can then be considered as interchanging cyclically only after a concept of cyclic interchange is established at group 1,2,3.

If I set the statuses or condition markers for each cyclic interchange in a table form

1	2	3
2	3	1
3	1	2

Hopefully you can see that each column advances around a circle coordinate system by a modulo(3)+1 addition rule which  is technically a cofactor of the modulo (3) addition .

At this level it is relatively easy to specify each condition marker( row in the table) and so to set up a "running into against set " table , that is "a running set in reverse" format.

Now if we compare this with the everyway toutable for exchange table , the permutation table we will find that interleaves the 2 tables , but we get to see that the group 1,2 flips before each cyclic interchange.

This is when we are free to rename that flip as also a cyclic interchange for "2 elements". Then instead of being locked into sign change as a negative or opposite or contra of the other we can see all sign changes as versions of cyclic interchange, as different kinds and magnitudes of rotation.

Rotation is thereby made the most fundamental grounds for the concept of a variables Foresign   at and in any step space,n-step/ rank/ stage/ level space and thus any descriptive Matrix of such a space .

Fixing the – sign as a half turn is thus unnatural and restrictive and explains why i takes on this role  of circular interchange. However we miss the point if we eulogise the invention of i, because it is the coordination of the circle which is the key. The clock arithmetics naturally reveal i to be an arc length marker, but one of many we are free to design.

[jehovajah]

Having now this insight we can perhaps make the design decision that the general sign simply should be .

Consider then the quaternion written as

You will see that this is not too different from Hermanns representation. The difference is merely in notation, but the geometry is radically different! is a circular arc segment is an oriented line segment .

Any combination of these 2 symbols or signs my be used to design general trochoidal segments( spiral filaments, plume filament segments, vortex filament segments etc). I will let you figure out what   you would use for i and what for –.

Now notationlly it really is insignificant if the markers are written on the line or as subscripts or super scripts, because it is their behaviour that is important. However these differences do allow a designer to introduce another more complex behavioural pattern. Thus in a power term aⁿ allows a  to obey one rule of composition or way of producting governed by the superscript. Thus a is subjugate to the superscript in a more complex way. But now the superscript element n itself can obey a different rule of composition or producting.

And we can take a greater step back to an even more general view and realise we do not have a general process of producting  or producing manufacturing that is more complex than knitting things together! There is no defineable multiplication at this generality. It reeifies as we specify addition and direction of addition, and it involves subjugation relationships. So it reeifies as a designable process.

This is the process I have called product design after an idea found in the general Doctrine of the Thought pattern.

As part of that design process the cycles or cyclic interchange group 1,2,3 plays a fundamental role.

We have looked at the rule of everyway unioning of factors. This is more usually called the Asdocative rule. Using this rule for 3( nb. 3) factors Hermann established the design of the Middler product. We know that 3 is necessary and sufficient to distinguish the cyclic interchange concept.

Using this concept of Asdociativity of factors the final design choice is how the concordance is evaluated. It can be evaluated by specifying the condition statuses that satisfy the definition , or it can be evaluated by the numeral result set out by numeral factor tables, the numerical resultants.

Simply, but also extremely impoverished in understanding , if they all give the same numeral resultants then every unioning or associating of the factors, and the applicable rules of subjugation upholds the rule of associativity which is the cyclus or cyclic interchange. Note however how the difference in the cycles , the running into against format is ignored usually by doing this!

But we know of the cyclic interchange of 2 factors . This is so clearly different to all 3 and greater cyclic interchange groups that it has been given its own set of rules called commutativity.

Commutativity is a form of cyclic interchange, so it is a mistake to set it as the foundation of factor interchange, and it is definitely a impoverishment of thought to define commutativity in terms of equal resultants.

[jehovajah]

I have written a lot on the origin of the function concept, but here we may now take Hermanns perspective.

Any product design is a function design, because a function is an expression with a fixed process related to it and contingent restraints or constraints.

The naturalistic design of a function is why it is so difficult to pin down. It is not a specific form, but a specifying process that designs a pattern of thought and behaviour. The cyclus group can be described in functional terms. This description is equivalent to attaching a label to some notion or process, or expression.

Since it is the same as this more general behaviour it really conveys no notational advantage over this labelling process. It is merely one style among many of labelling the complex thought patterns we design to produce synthesis results or even analytical ones.

The person who is at the base of Hermanns approach is Justus his father. His paper on the Naturalistic approach to science and Mathematics is worth a read.

[jehovajah]

Today, on this Passover day, the first day of Chmetz, having onsidere Ezra, the founder of Judaim under Cyrus the great Messiah it is fitting to consider how this scribal functionary performed a duty of Statecraft for the emperor which took decades to accomplish! But he was chosen for his steadfast loyalty to the emperor and his cause, so it is that a small dedicated group can become a Manipulation of a leader, an extension of his body and his will.

These essential ideas behind the mathematical concept of a function are redolent in Hermanns synthetical treatment by labels. It is important that labels, once assigned a function, retain hat precise function. Therefore it is vital that one design the definition with expertise, and not carelessly!

This is why Hermann focuses so much on product design.D
Perhaps the designing of products in mathematics so called is deemphasised, but it is key to Hetmanns doctrine of the Thought Process.

I want to compare 2 product designs for labelling rotation, one by Euler, the other by Norman Wildberger .

Firstly they both use the now conventional function format, but they are labels of rather complex expressions?

exp(ix)= cosx+ isinx where cosx and sins are infinite power series expansions in x

And

e(u:t) u²-t²/u²+t²:2ut/u² + t² := u²-t²/u²+t²+i2ut/u² + t²

Both are based on the circle but Euler uses a direct parameter, the arc length x of the circle, while Norman uses a parameter on the y axis called t and insists on rational values Only.

I have written them both as combinations of products to bring them in line with The representation of Extending magnitudes Hemann uses. Both can be written as ordered pairs , for graphing purposes.

Now Gauss is rightly credited with this combination format for complex numbers, but we may thank Justus Grassmann for the representation of 3 or more such terms . His papers and work, within a small group of Crystallographers are largely unrecognised even today by Grassmann scholars.

The Ausdehnungsgröße have a clear origin in his work, while Gauss provided a glimpse of the power of this notational device or labelling scheme.

Euler, in designing his product, wanted to focuss on the arc length, and thus employed the radius measure for the arc( the Halbmesser) otherwise known as the Radian. While it is not definitely clear we can link this back to Sir Roger Cotes as a Harmonium Mensurarsm, a way of harmonising all measurements.

Due to Newton and then Leibnitz , the Taylor expansion of the sine line segments as the arc increased was shown to be a binomial series expansion of an infinite nature. The cosine, the logarithm all followed suit. But it was Newton that resolved the arcane expansion.

These infinite sums were called series, and we're always considered pragmatically as approximations, truncated where needed. They were well conceived abd designed by Newton who had no need for convergence theorems because he did not believe in infinite processes.

Because these derived from the binomial expansion , they were understood as Products. They were renamed series by some who ignored the binomial series product from which they derive. In fact Leibniz did not and could not derive the binomial series expansion, as is recorded in a letter where he asked Newton how he came to derive it, and thus his Fluxions.

While it may seem of academic interest, the Fluxions are based on a radically different principle to Leibniz differentials. They are based on this binomial "series" expansion and thus are always well firmed, and constrained by the priduct. Without this product design the Fluxions would have been flawed, as was Leibniz conception, a poit remarked upon by Cotes in his preamble to the Astrologers Principles.

Infinitesimals are a fanciful but not well conceived notion despite the so called limit process.

So the binomial series product design underpins the Euler formula for the rotation around the arc. The values are the long establishes sine and cosine table values, which are usuall written in regard to the amount of angle turned. By changing the angle in degrees to an angle in arc lengths(! Precisely the same idea!) and straightening the circle out into a " straight" line using a /2 scale transform we get the products as binomial expansions.

In the course of performing this feat Newton figured out the combinatorial formula for the terms, linking this product design to the combinatorial doctrine explored by Galois et al.

Norman designed his product to be rational, and used the same Pythagorean triangle relationships, but this time he does not base the triangle from the centre ! Because of this he does not focus on the arc length, but rather the tangent length that passes through the circle centre, in a right angled triangle whose " origin is at -1, or one end of the diameter. This utilises a very short binomial Square product process. The difficulty is what are we labelling? We can answer that using the Ausdehnungsgröße labelling, in a straight forward way, whereas we really are dependent on the diagram, and the Cartesian coordinate frame to even begin to describe what we are doing.

In this particular instance Hermann is writing the expression in terms of Homogenous coordinates, and that requires a prior theory of homogenous coordinates. fortunately Norman has one in his Universal hyperbolic Geometry.

However we can see that if we set out the extended magnitude product as a binomial type expansion, then we can quickly set the resultant as a closed result by square products from hyperbolic ones , and then setting the form as some numeral plus some multiple of t.

Because Norman ultimately bases his forms on the type Nat( natural numbers) he is able to do an extra bit of manipulation changing " variables"(?) into numbers where it suits. So the labels have a different meaning within the function bracket , there they are variables. On the other side of the expression there they have to be specific " numbers" passed through to them by the variable memory blocks.

Wolfram Alpha is different. It evaluates everything as an expression. This very powerful idea is found in Grassmanns work, but really takes some getting used to.

The product design comes first, then the elements that satisfy that design in the Assiciative case come next.

This is what we will see demonstrated in the translation of page 379

[jehovajah]

Additional sub and super marks (subscripts, superscripts) give labels extra expressive qualities on the page , when designing a product . This is the product label design aspect that makes a notation helpful or a hindrance.

The use of exponents is a simple notation that directly links two related but Differring knitting processes. Subjugation of synthetic knitting( addition) underpins the concept of a multiplication. But few realise that it does not define multiplication.

The concept of multiplication is really undefined in most processes, and yet we think we know what it is. I have explored this issue in detail so I am only referencing it to highlight Exponentiation as a way of counting like factors . And because it is a direct count, we use a synthetic knitting approach. Thus the superscripts behave like a synthetic knitting. Consequently we can place labels for line segments in the superscript position, if we wish to design a product which makes sense of that.

The isolation of the base from the superscript means we can draw up a table for the base knitting and a table for the superscript knitting . The base knitting if subjugating can then be constrained by the superscript knitting.

We can see this aspect of label design in the exponential function . The exponential label services it's usefulness from the work done on Logarithms by Briggs , but took its final shape in Wallis manuscript on Algebra, in which he put forward many pouch label designs..

In Hermanns and Hamiltons time, function theory was only just beginning, and really took its authority from Cantirs set Notation. Prior to that an expression made use of subscripts or superscripts to identify the arguments or dependents in a process. Eulers superscript design for the exponential function was thus very astute, but confusing if read as an equality rather than an identifying label!

Today we still have that naive response to the function design thinking it is an equation rather than an identifying label.

In that regard the closing in product design is usually set = to the tally mark 1 or generally cosø where ø is an arc segment of the unit circle that cyclically rotates one line segment onto anther ( not just straight line segments! ). All our numerals are in fact tabulated in our sine tables in principle. Dedekinds notion of the number line, based on Wallis's notion of the Measuring line typically ignores this sine tabular structure. Instead an endless straight line is posited as the standard. This is not unusual, and in fact Hermann uses it as his foundational system for line segments.

However in the Doctrine of the though processes it is clear this is a formal construct , not a reality. The gometric fundamentals are pragmatic line segments, and these exist within a spherical space as an n- th step system. Utilising the tally marks within the sine tables is making a powerful statement: numbers are not real! They are accounting signs applicable anywhere and in any orientation.

So while I promote [e|e]= 1e because it retains all the information of the vertical projection result I also recognise that in a system that is subjugating, that projection occurs throughout the system, because that verticl projection line has parallels that occur throughout the system. And while we may be identifying a specific orientation the usefulness of that product in geometry is in the tally mark it identifies .

But I perceive that in physical space many forces when there components are projected onto a particular orientation do produce an identifiable resultant in that orientation or rather the plane of that orientation. For that reason I promote the fuller version, to which we may then add further constraints as needed.

This is a design preference. When I designed the Newtonian triples I made this choice, but in order to render it I had to add the further constraint for quaternions that n the Newtonian line segment would be represented by the numeral or scalar part of the Quaternion.

Although I clearly multiplied 2 Newtonian triples to render the Mandelbrot sculpture, I was surprised by the cubed result. However I never even thought to develop the product by means of the associativity principle as Hermann has done here!

[jehovajah]

I also acknowledge the direct link between [e|e] and katameetresee the Greek notion for both counting and measuring!. In this activity we formally lay a lesser magnitude(quantity later!) down onto a a greater magnitude. This is the experience of Monas! We say "1" or "one", Monas is this singling out experience.

Technically then 1 is not a number. It is a unit Metron, a Monas. The first number or quantity is thus 2.

The shadow product is our act of fractalising magnitude into quantity, in a regular fractal pattern. While it derives out of directed line segments, oriented projections of Metrons of all forms or conceptions, we typically abstract the tally count. In so doing we acknowledge division as the principal source of our concept of number, thus all knitting is subjugate to some initial division or fractal process.

To then order these knittings in terms of subjugation it makes sense to finalise the order with division as an analytical subjugation : that is division is that analytical search for the factors of a subjugate knitting, a product in which a former " representation" is transformed by a higher step creating element..

The colliding in product recognises our fractal processing( but here we deliberately keep it uniform) and the out spreading product recognises our space extending ability( while also including our rotational extending ability too!). We measure in any direction , count in any orientation, and at the same time create or recognise forms in any direction, orientation or rotating dynamic.

[jehovajah]

Normannmaking the point about the fundamental role of the closing in product.

<a href="http://www.youtube.com/v/yAb12PWrhV0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yAb12PWrhV0&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/yAb12PWrhV0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yAb12PWrhV0&rel=1&fs=1&hd=1</a>

Notice how the parallel lines projected in this way form the subjugate forms of the spreading out product: the parallel sides of the parallelograms are the factors of the spreading out product. The parallelogram is the output result of the spreading out product for each of those specified line segments.

Clearly how we orient the factors determines the orientation of the parallelogram. I have discussed how interchanging the factors cyclically alters the orientation of the parallelogram such that it's rotation marker switches between